Title: Linear Algebra
Tags: midterm chapter1.2 chapter1.5 chapter1.1 chapter1.3 chapter1.4

Q: A [...] takes the form [...].
A: linear equation; \(a_1x_1 + a_2x_2 + ... + a_nx_n = b\)

Q: A [...] is a set of one or more [...].
A: system of linear equations (SLE); linear equations
Extra: Usually arranged by variable.
Example:
\[4x_1 - 7x_2 +6x_3 = 5 \\
1x_1 - 0x_2 +9x_3 = 3\]

Q: [...] takes the form \(m \text{ x } n\), with \(m\) representing rows and \(n\) representing columns.
A: Matrix notation

Q: What is an augmented matrix?
A: An augmented matrix represents an entire system of linear equations, including constants (and equal sign).
\[
\left[\begin{array}{@{}ccc|c@{}}
1 & -2 & 1 & 0 \\
0 & 2 & -8 & 8 \\
-4 & 5 & 9 & -9
\end{array}\right]
\]
The vertical line denotes the equal sign and is not always included.

Q: What is a coefficient matrix?
A: An coefficient matrix represents a system of linear equations, excluding constants.
\[\begin{bmatrix}1 & -2 & 1 \\0 & 2 & -8 \\-4 & 5 & 9\end{bmatrix}\]

Q: [...] SLEs have [...].
A: Equivalent; the same solution set

Q: What are the three elementary row operations?
A: Replacement: replacing one row with its sum with another.
Interchange: swapping two rows.
Scaling: multipling an entire row by a scalar.

Q: The [...] of an SLE is the set of [...].
A: solution set; all possible solutions
Extra: There are either 0, 1, or infinite solutions.

Q: A [...] to a SLE is [...].
A: solution; a list of numbers that solve each equation simultaneously
Extra: \((s_1, s_2, ..., s_n)\)

Q: A [...] is the [...].
A: pivot; leftmost non-zero entry in a row
Extra: Also known as the leading entry.

Q: A [...] is an SLE with [...] solutions.
A: consistent system; one or infinite

Q: An [...] is an SLE with [...].
A: inconsistent system; no solution
Extra: Inconsistent systems exist when an augmented matrix takes the following form: \[\left[\begin{array}{@{}cc|c@{}}... & ... & ... \\0 & 0 & g\end{array}\right]\]
Where \(g\) is some non-zero number.

Q: What steps will solve a system of linear equations?
A: 1. Use row operations to replace the system with an equivalent system that's easier to solve.
2. Repeat.

Q: When is a matrix in reduced row echelon (RREF) form?
A: 1. All rows of zeros are at the bottom.
2. All entries above and below pivots are zero.
3. Each pivot is 1.
\[\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}\]

Q: When is a matrix in row echelon (REF) form?
A: 1. All rows of zeros are at the bottom.
2. All entries below a pivot or leading entry are zero.
\[
\begin{bmatrix}
1 & -2 & 1 \\
0 & 2 & -8 \\
0 & 0 & 9 \\
0 & 0 & 0
\end{bmatrix}
\]

Q: A [...] in a matrix is the location of a [...] in a row-echelon form of the matrix.
A: pivot position; pivot

Q: A [...] is a column that contains a [...].
A: pivot column; pivot position

Q: A [...] corresponds to a [...].
A: non-pivot column; free variable

Q: If the reduced row-echelon form of an equation has a row of the form \([000...b]\), where \(b\) is some non-zero number, then the matrix is [...] and has [...].
A: inconsistent; no solution

Q: If the reduced row-echelon form of an augmented matrix has a pivot in every column (besides the augmented column), then there exists [...].
A: one (unique) solution
Extra: This also means that there will be ones down the diagonal, and that the rightmost column will contain the values of each corresponding variable.

Q: If the reduced row-echelon form of an equation contains [...], then there exist [...].
A: free variables; infinite solutions
Extra: Non-pivot columns represent free variables, also known as parameters.

Q: [...] are arrows beginning at [...].
A: Vectors; the origin

Q: Algebraically, [...] are just matrices with [...] column.
A: vectors; one

Q: A [...] system of linear equations is one that can be written in the form [...].
A: homogenous; \(A\vec{x}=\vec{0}\)
Extra: The trivial solution is just \(\vec{0}\).

Q: The homogenous equation \(A\vec{x}=\vec{0}\) has only the trivial solution (\(\vec{0}\)) if [...] is a pivot column.
A: every column

Q: The homogenous equation \(A\vec{x}=\vec{0}\) has a non-trivial solution if [...] is not a pivot column.
A: at least one column

Q: A [...] system of linear equations is one that can be written in the form [...].
A: non-homogenous; \(A\vec{x}=\vec{b}\)

Q: A [...] takes the form [...] where \(\vec{v}\) represents vectors and \(c\) represents scalars.
A: linear combination; \[c_1\vec{v}_1 + c_2\vec{v}_2 + ... + c_p\vec{v}_p\]

Q: [...] is the [...] of \(\vec{v}_1, ... , \vec{v}_p\).
A: Span; set of all linear combinations
Extra: It is denoted \(\text{Span\{}\vec{v}_1, ... , \vec{v}_p\text{\}}\).

Q: In general, \(\text{Span\{}\vec{u}\text{\}}\) is a [...] through the [...].
A: line; origin

Q: In general, [...] is a [...].
A: \(\text{Span}\{\vec{u},\vec{v}\}\); plane through the origin

Q: A [...] takes the form:
\[\begin{bmatrix}1 & 2 & -1 \\ 0 & -5 & 3\end{bmatrix}\cdot\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix}4\\1\end{bmatrix}\]
A: matrix equation
Extra: Wherein the first matrix is \(A\), the second matrix is \(\vec{x}\), and the third matrix is \(\vec{b}\).

Q: A [...] takes the form:
[...]
A: vector equation; \[x_1\cdot\begin{bmatrix}1\\0\end{bmatrix}
+x_2\cdot\begin{bmatrix}2\\5\end{bmatrix}
=\begin{bmatrix}4\\1\end{bmatrix}\]

Q: A set of vectors is linearly [...] if removing one vector [...] their span.
A: dependent; does not reduce

Q: The [...] of a set of vectors is the set of all possible [...] of those vectors.
A: span; linear combinations

Q: If each vector added to a set adds another [...] to their span, then the set of vectors is [...].
A: dimension; linearly independent

Q: How do you find all vectors that are mapped to the zero vector?
A: 1. Create augmented matrix with rightmost column equal to 0.
2. Solve for \(x_1\), \(x_2\), etc.

Q: How do you find out if \(\vec{b}\) is in the range of the transformation \(\vec{x} \rightarrow A\vec{x}\)?
A: 1. Create augmented matrix with rightmost column equal to \(\vec{b}\).
2. Solve for \(x_1\), \(x_2\), etc.
3. If consistent, it is in range. If inconsistent, it is not in range.

Q: Every matrix transformation is a linear transformation. True or false?
A: True.

Q: If \(A\) is an \(m \text{ x } n\) matrix, then the range of the transformation \(\vec{x} \rightarrow A\vec{x}\) is [...].
A: \(\mathbb{R}^m\)

Q: A linear transformation preserves the operations of [...] and [...].
A: vector addition; scalar multiplication

Q: A linear transformation is a special type of [...].
A: function

Q: A linear transformation \(\mathbb{R}^n \rightarrow \mathbb{R}^m\) always maps the origin of [...] to the origin of [...].
A: \(\mathbb{R}^n\); \(\mathbb{R}^m\)

Q: Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^4\) be a linear transformation with
\[T(\vec{e_1}) = \begin{bmatrix} -6 \\ -3 \end{bmatrix}\]
and
\[T(\vec{e_2}) = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\]
The standard matrix for \(T\) is [...]
A: \[\begin{bmatrix} -6 & 2 \\ -3 & -4 \end{bmatrix}\]

Q: Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be a vertical shear transformation that maps \(\vec{e_1}\) to \(\vec{e_1} + 4\vec{e_2}\) but leaves \(\vec{e_2}\) unchanged. Then the standard matrix for \(T\) is:
[...]
A: \[
\begin{bmatrix}
1 & 0\\
4 & 1
\end{bmatrix}
\]

Q: Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be a transformation that rotates a vector \(\theta\) radians counterclockwise about the origin. The standard matrix \(T\) is:
[...]
A: \[\begin{bmatrix}
\cos{\theta}&-\sin{\theta}\\
\sin{\theta}&\cos{\theta}
\end{bmatrix}\]

Q: If \(A\) has a [...] in every row, then the columns of \(A\) [...] \(\mathbb{R}^m\), and for each \(\vec{b}\) in \(\mathbb{R}^m\), the equation \(A\vec{x}=\vec{b}\) has a [...].
A: pivot; span; solution

Q: If \(A\) has more rows than columns, then there cannot be a pivot in every row, and \(A\) [does/does not] span \(\mathbb{R}\).
A: does not

Q: When a matrix \(A\) has a pivot in every column, then its solution is [...].
A: unique

Q: Let the multiplication of two matrices \(A\) and \(B\), be defined by \((Q \text{ x } R)(S \text{ x } T)\). Then [...] must equal [...] and the resulting matrix will have the dimensions [...].
A: \(R\); \(S\); \((Q \text{ x } T)\)

Q: A [...] equation can be written in the form [...].
A: homogeneous; \(A\vec{x}=\vec{0}\)

Q: A matrix must be [...] to be invertible.
A: square

Q: If a matrix \(A\) is invertible, then each column contains a [...].
A: pivot

Q: The inverse of a matrix [always/doesn't always] exist. It [...] exists for a non-square matrix.
A: doesn't always; never

Q: \(T^{-1}(T(\vec{x}))\) = [...]
A: \(\vec{x}\)

Q: The [...] can be described as the [...] matrix, because it returns the same matrix.
A: Identity matrix \(I\); "do-nothing"

Q: If a transformation \(T\) collapses space, then \(T\) has [...].
A: no inverse

Q: If a transformation \(T\) has more columns than rows, then \(T\) has [...].
A: no inverse

Q: If a transformation \(T\) has more rows than columns, then \(T\) has [...].
A: no inverse

Q: A subspace of \(\mathbb{R}^2\) is any set \(H\) that satisfies which properties?
A: - \(H\) passes through the origin
- \(H\) contains the sums of all vectors in \(H\)
- \(H\) contains all scaled vectors in \(H\)
In other words, the subspace passes through the origin and goes on forever.

Q: What are the subspaces of \(\mathbb{R}^3\)?
A: - Point (at origin)
- Line (through origin)
- Plane (through origin)
- Whole space

Q: The column space of a matrix \(A\), Col A, is the [...] of the columns of \(A\). This is also known as the [...] of A.
A: span; range

Q: Null space is a subspace of the [...].
A: domain

Q: Col space is a subspace of the [...].
A: codomain

Q: Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be a transformation that reflects a vector across the line \(y=x\). The standard matrix \(T\) is:
[...]
A: \[\begin{bmatrix}
0&1\\
1&0
\end{bmatrix}\]

Q: Let \(T(x,y,z)=(−x−y+2z,−5x−y−9z)\). What is the standard matrix for \(T\)?
A: \[
\begin{bmatrix}
-1 & -1 & 2 \\
-5 & -1 & -9
\end{bmatrix}
\]

Q: A transformation \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) is [...] if, for every vector \(b\) in \(\mathbb{R}^m\), the equation \(T(x)=b\) has [...] solution \(x\) in \(\mathbb{R}^n\).
A: one-to-one; at most one

Q: A transformation \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\) is [...] if, for every vector \(b\) in \(\mathbb{R}^m\), the equation \(T(x)=b\) has [...] solution \(x\) in \(\mathbb{R}^n\).
A: onto; at least one
Extra: Central question: Is the range the entire codomain? If yes, then the mapping is onto.

Q: The vector \(\vec{v}=\begin{bmatrix}4\\10\\6\end{bmatrix}\) [is/is not] in Nul\(A\), where  \(A=\begin{bmatrix}-2&-1&3\\4&-1&-1\\0&0&-3\end{bmatrix}\).
A: is not
Extra: In order to be in Nul\(A\), the matrix \(A\) and vector \(\vec{v}\) must multiply to the zero vector.

Q: The [...] of a matrix \(A\) is the set of all vectors which, when multiplied with \(A\), equal [...].
A: null space; the zero vector

Q: The [...] of a matrix \(A\) is the set of all linear combinations of [...]. This is also known as the [...] of those vectors.
A: column space; the column vectors of \(A\); span

Q: Given the equation \(A\vec{x}=\vec{b}\), [...] are given and one must solve for [...].
A: \(A\) and \(\vec{b}\); \(\vec{x}\)

Q: Given the equation \(A\vec{x}=\vec{b}\), [...] is the [...] of the system of equations.
A: \(A\); coefficient matrix

Q: What can be deduced from a matrix \(A\) that has more rows than columns?
A: - There is not a pivot in every row.
- For some \(\vec{b}\) in \(\mathbb{R}^m\), \(A\vec{x}=\vec{b}\) has no solution.
- The columns of \(A\) do not span \(\mathbb{R}^m\).

Q: The [...] is denoted [...]. Also known as the [...].
A: zero vector; \(\vec{0}\); trivial solution

Q: When a set of vectors is [...], then the equation \(A\vec{x}=\vec{0}\) has [...].
A: linearly dependent; more than the trivial solution

Q: A set of linearly independent vectors have [...].
A: maximum span

Q: The maximum number of linearly independent vectors in \(\mathbb{R}^n\) is [...].
A: n

Q: A pivot in every row means that the linear system \(A\vec{x}=\vec{b}\) has [...] solution for every \(b\).
A: at least one

Q: A pivot in every column means that the linear system \(A\vec{x}=\vec{b}\) has [...] solution for every \(b\).
A: at most one

Q: A pivot in every row and a pivot in every column (known as "full rank") means that the linear system \(A\vec{x}=\vec{b}\) has [...] solution for every \(b\).
A: exactly one

Q: If a set of vectors contains more vectors than entries, then the set is linearly [...].
A: dependent

Q: If a set of vectors contains the zero vector (\(\vec{0}\)), then the set is linearly [...].
A: dependent

Q: In general, columns in a matrix \(A\) denote [...] and rows denote [...].
A: vectors; their entries

Q: Two matrices can be added together. True or false?
A: True.

Q: A matrix can be multiplied by a scalar. True or false?
A: True.

Q: What is the additive inverse of \(\begin{bmatrix}-1&4\\-3&4\end{bmatrix}\)?
A: \(\begin{bmatrix}1&-4\\3&-4\end{bmatrix}\)

Q: \[
\begin{bmatrix}
2&3\\
1&-5
\end{bmatrix}
\cdot
\begin{bmatrix}
4\\
1
\end{bmatrix}
\]
=
[...]
=
\[
\begin{bmatrix}
11\\
-1
\end{bmatrix}
\]
A: \[4\cdot
\begin{bmatrix}
2\\
1
\end{bmatrix}
+
1\cdot
\begin{bmatrix}
3\\
-5
\end{bmatrix}
\]

Q: The [...] has 1's down the left-to-right diagonal and 0's elsewhere: \[\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\]
A: identity matrix (\(I\))
Extra: Multiplying a matrix \(A\) by the Identity matrix \(I\) equals the original matrix: \(A\times I = A\)

Q: Does \(A \times B=0\) imply that \(A=0\) or \(B=0\)?
A: No.

Q: Does \(A^k=0\) imply \(A=0\)?
A: No.

Q: Given a matrix \(A\) equal to \(\begin{bmatrix}1&2&3\\1&-1&0\end{bmatrix}\) then its transpose, \(A^T\), is:
[...]
A: \[\begin{bmatrix}1&1\\2&-1\\3&0\end{bmatrix} \]
Extra: The columns of \(A^T\) are formed from the corresponding rows of \(A\).

Q: \(A^{T^T}=\)[...]
A: \(A\)

Q: An \(n \times n\) matrix \(A\) is [...] if there is a \(n \times n\) matrix \(C\) such that [...].
A: invertible; \(AC=I\) and \(CA=I\)

Q: Let \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\). If [...], then \(A\) is [...] and:
\[A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\]
A: \(ad-bc\neq0\); invertible

Q: If \(A\) is an invertible \(n\times n\) matrix, then for each [...] in \(\mathbb{R}^n\), the equation \(A\vec{x}=\vec{b}\) has the unique solution [...].
A: \(\vec{b}\); \(\vec{x}=A^{-1}\vec{b}\)
Extra: In other words, multiplying the inverse of \(A\) by \(\vec{b}\) gives \(\vec{x}\).

Q: \((A^{-1})^{-1}=[...]\)
A: A

Q: \( (AB)^{-1}= [...] \)
A: B^{-1} A^{-1}
Extra: Order matters!

Q: \((A^T)^{-1}=\)[...]
A: \((A^{-1})^T\)
Extra: Order matters!

Q: A matrix \(A\) is [...] if and only if it is convertible to [...].
A: invertible; \(I\)

Q: How is the inverse of a matrix found?
A: 1. Form the augmented matrix [\(A\)|\(I\)].
2. Row-reduce the matrix until the left side becomes \(I\). Then the right side is \(A^{-1}\).
3. If \(I\) cannot be made from the left hand side, then the inverse of \(A\) does not exist and \(A\) is not invertible.
To check work:
\(A\times A^{-1}= I\)

Q: If \(A\) is row-equivalent to \(I\), then there must exist [...] which transform \(A\) to \(I\), which means that \(A\) has pivots in every [...], which means that each column is linearly [...].
A: row-operations; column; indepenent

Q: A [...] \(T\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a rule that assigns to each vector \(\vec{x}\) in \(\mathbb{R}^n\) a vector \(T(\vec{x})\) in \(\mathbb{R}^m\).
A: transformation
Extra: A transformation is also known as a function, or mapping.

Q: The space that a matrix \(A\) inhabits is known as its [...].
A: domain

Q: The space that a transformed matrix \(T(A)\) inhabits is known as its [...].
A: codomain

Q: The set of all outputs of a transformation is known as its [...].
A: range
Extra: - The range of \(T(A)\) is always a subset of the codomain.
- The range of \(T(A)\) is sometimes equal to the codomain.

Q: Given a matrix \(A=\begin{bmatrix}1&-1\\3&5\\-1&7\end{bmatrix}\) used for a transformation, the dimensions of the input must be [...] and the dimensions of the output must be [...].
A: \(2\times 1\); \(3 \times 1\)

Q: Given transformation matrix defined \(T : \mathbb{R}^n \rightarrow \mathbb{R}^m\), \(n\) corresponds to the number of [...] in that matrix and \(m\) corresponds to the number of [...] in that matrix.
A: columns; rows

Q: Given a matrix \(A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}\), what geometric operation does it perform?
A: It "erases" information from the 3rd dimension. Every vector is mapped into the \(xy\) plane in \(\mathbb{R}^3\).

Q: The matrix [...] will reflect across the [...] axis.
A: \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\); \(x\)

Q: The matrix [...] will reflect across the [...] axis.
A: \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\); \(y\)

Q: The matrix [...] will reflect across the [...].
A: \(\begin{bmatrix}-1&0\\0&-1\end{bmatrix}\); origin

Q: The matrix [...] will reflect across the [...].
A: \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\); line \(y = x\)

Q: The matrix [...] will [...].
A: \(\begin{bmatrix}1&f\\0&1\end{bmatrix}\); shear the output on the \(x\) axis

Q: The matrix [...] will [...].
A: \(\begin{bmatrix}1&0\\f&1\end{bmatrix}\); shear the output on the \(y\) axis

Q: \(T(\vec{0})=\) [...]
A: \(\vec{0} \)
Extra: In other words, a transformation will always map the origin to the origin.

Q: \(T(c\vec{v})\)=[...]
A: \(cT(\vec{v}) \)
Extra: For example, the negative will go where we want it to (in the opposite direction).

Q: \(T(c\vec{u} + d\vec{v})=[...]\)
A: cT(\vec{u}) + dT(\vec{v})

Q: Linear transformations preserve [...] and [...].
A: the origin; straightness

Q: If we have a set that spans the domain, then knowing where \(T\) takes that set tells us what \(T\) does to [...].
A: every vector

Q: Knowing what a transformation \(T\) does to [...] tells you what happens to [...].
A: the identity matrix; every vector

Q: The columns of [...] are the images of the columns of \(I_n\).
A: the transformation matrix

Q: If a mapping's range is equal to its codomain, then the mapping is [...].
A: onto
Extra: Central question: Is the range the entire codomain? If yes, then the mapping is onto.

Q: If a mapping is [...], then everything in the codomain is [...] by something in the domain.
A: onto; mapped to (or "hit")
Extra: Central question: Is the range the entire codomain? If yes, then the mapping is onto.

Q: If a mapping collapses space, then it [is/is not] one to one.
A: is not

Q: If a mapping drops dimensions, then it [is/is not] one to one.
A: is not

Q: The [...] that spans a subspace is the [...] for that subspace.
A: minimum set of vectors; basis

Q: The [...] of Nul\(A\) is equal to the number of [...] in [...].
A: dimension; non-pivot columns; RREF(\(A\))

Q: The [...] of Col\(A\) is equal to the number of [...] in [...].
A: dimension; pivot columns; RREF(\(A\))

Q: \(\mathbb{R}^2\) is a two-dimensional subspace of \(\mathbb{R}^3\). True or false?
A: False.

Q: Given a matrix: \[A=\begin{bmatrix}1&-5&7&0\\1&-1&0&0\\-2&10&-14&0\end{bmatrix}\]
with the RREF of:
\[A=\begin{bmatrix}1&0&-7&0\\0&1&-7&0\\0&0&0&0\end{bmatrix}\]
What is the column basis of Row(\(A\))?
A: Row(A) = {\(\begin{bmatrix}1\\0\\-7\\0\end{bmatrix}\),\(\begin{bmatrix}0\\1\\-7\\0\end{bmatrix}\)}
Steps:
1. Convert original matrix \(A\) to RREF.
2. Identify rows with pivots.
3. Row(\(A\)) is simply these rows transformed into vectors.

Q: Given an augmented matrix: \[A=\begin{bmatrix}1&1&1&1&0\\1&2&3&4&0\\4&3&2&1&0\end{bmatrix}\]
with the RREF of:
\[A=\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\\0&0&0&0&0\end{bmatrix}\]
What is the column basis of Nul(\(A\))?
A: Nul(A) = Span{\(\begin{bmatrix}1\\-2\\1\\0\end{bmatrix}\),\(\begin{bmatrix}2\\-3\\0\\1\end{bmatrix}\)}
Central question: Which set of vectors, when multiplied by \(A\), equal 0?
Steps:
1. Form augmented matrix from \(A\) (if not already augmented).
2. Reduce to RREF.
3. Solve.
4. Convert to a vector equation.

Q: Given a matrix: \[A=\begin{bmatrix}1&-5&7&0\\1&-1&0&0\\-2&10&-14&0\end{bmatrix}\]
with the RREF of:
\[A=\begin{bmatrix}1&0&-7&0\\0&1&-7&0\\0&0&0&0\end{bmatrix}\]
What is the column basis of Col(\(A\))?
A: Col(A) = {\(\begin{bmatrix}1\\1\\-2\end{bmatrix}\),\(\begin{bmatrix}-5\\-1\\10\end{bmatrix}\)}
Steps:
1. Convert original matrix \(A\) to RREF.
2. Identify columns with pivots.
3. Col(\(A\)) is equal to the columns from the original matrix which are pivot columns in the RREF matrix.

Q: The dimension of the null space of \(A\) is the number of columns of \(A\) that [are/are not] pivot columns.
A: are not

Q: The [...] of \(A\) is the same as the [...] of \(A^T\).
A: row space; column space

Q: The [...] of the row space of \(A\) and the column space of \(A\) are the same.
A: dimensions

Q: If \(A\) and \(B\) are row equivalent, then their [...] are the same.
A: row spaces

Q: A [...] is a [...] that is as [...] as possible.
A: basis; spanning set; small

Q: The [...] of a nonzero subspace \(H\), denoted [...] is the number of [...] in any basis for \(H\).
A: dimension; dim\(H\); vectors

The rank of a matrix \(A\), denoted rank \(A\), is the dimension of the column space of \(A\). It's always equal to the number of pivot columns in \(A\).

Q: The Rank-Nullity Theorem states that [...] plus [...] equals the number of [...] in \(A\).
A: rank \(A\); dim Nul \(A\); columns

Q: Suppose a 3 x 4 matrix \(A\) and rank(\(A\)) = 2. What is dim Nul \(A\)? What and how does it transform?
A: dim Nul = 2. The matrix \(A\) transforms a vector in \(\mathbb{R}^4\) into a 2D subspace of \(\mathbb{R}^3\), collapsing 2 dimensions along the way.

Q: If the determinate of a matrix \(A\) is [...], then the matrix [is/is not] invertible.
A: 0; is not

Q: If the determinate of a matrix \(A\) is [...], then the rows and columns of that matrix are linearly [dependent/independent].
A: 0; dependent
Extra: This relationship works both ways.

Q: Given a matrix:
\[A=\begin{bmatrix}1&-4&15\\-13&36&-11\\-4&12&-37\end{bmatrix} = \begin{bmatrix}2&1&8\\7&-9&-60\\2&-3&-20\end{bmatrix} \begin{bmatrix}1&0&0\\0&1&0\\0&0&4\end{bmatrix} \begin{bmatrix}0&1&-3\\-5&14&-44\\7&-2&6\end{bmatrix}\]
The eigenvalues of \(A\) are: [...]
A basis for the largest eigenvalue's eigenspace is: [...]
A basis for the smallest eigenvalue's eigenspace is: [...]
A: 1, 4; \[\begin{bmatrix}8\\-60\\-20\end{bmatrix}\]; \[\begin{bmatrix}2\\7\\2\end{bmatrix}, \begin{bmatrix}1\\-9\\-3\end{bmatrix}\]