Linear Algebra Exam 2
If A and B are n x n matrices, then set AB is equal to...
(det A) * (det B)
Let A be a square matrix, if two rows of A are interchanged to produce a matrix B, then det B equals
- det A
If the rank of a 9x8 matrix A is 7, what is the dimension of the solution space Ax=0
1 ( 8 - dimNulA = 7, dimNulA=1)
A subspace of Rⁿ is any set H in Rⁿ that has three properties
1) The zero vector is in H 2) For each u and v in H, the sum u + v is in H 3) For each u in H and each scalar c, the vector cu is in H
If the subspace of all solutions of Ax = 0 has a basis consisting of three vectors and if A is a 5 x 7 matrix, what is the rank of A
4 (the dimNulA = 3 as described and there are 7 columns. 7 - 3 = 4)
What is the rank of a 6x8 matrix whose null space is three-dimensional
5 (dimNulA = 3 and there are 8 columns. 8 - 3 = 5)
Linearly independent set in H that spans H for a subspace H of Rⁿ
Basis
if a matrix A is m x n, what integer value must p be so that Col A is a subspace of R^p
Col A is a subspace of Rⁿ since each column vector has n entries
The set Col A of all linear combinations of the columns of A
Column Space of A
A matrix has linearly independent vectors if its determinate...
Does not equal zero
A square matrix A is invertible if and only if its determinate...
Does not equal zero
How would you construct a 3x5 matrix A such that dimNulA = 3 and dimColA=2
Ensure it has exactly two pivot columns which creates three free columns
(True/False) If B is an echelon form of matrix A, then the pivot columns of B form a basis for Col A
False, B and A don't work this way
(True/False) det (A + B) = det A + det B
False, det AB = det A + det B
(True/False) The determinant of A is the product of the diagonal entries of A
False, only if the matrix is in triangular form
(True/False) A subspace of Rⁿ is any set H such that 1) The zero vector is in H 2) u, v, and u + v are in H 3) c is a scalar and cu is in H
False, parts 2 and 3 should state for each
(True/False) A subset H of Rⁿ is a subspace if the zero vector is in H
False, that is only one of three conditions
(True/False) The column space of matrix A is the set of solutions of Ax = b
False, the column space is all the vectors b for which there is a solution Ax = b
(True/False) If det A is zero, then two rows or two columns are the same, or a row or a column is zero
False, the described condition is not required in all cases
(True/False) The determinate of the transpose of A is equal to -det A
False, the determinant of the transpose of A is det A
(True/False) The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^m
False, the subspace would be Rⁿ
(True/False) Let H be a subspace of Rⁿ. If x is in H, and y is in Rⁿ, the x+y is in H.
False, x + y are in Rⁿ but y is not necessarily in H only x
Let A be an n x p matrix whose column space is p-dimensional. Explain why the columns of A must be linearly independent.
If the matrix is n x p that means there are p columns. If the column space is p as well, that means the rank is p and there are p pivot columns. If all the columns contain a pivot then the columns of A must be linearly independent.
In order for a set of vectors to form a basis for Rⁿ they must be
Invertible
if a matrix A is m x n, what integer value must q be so that Nul A is a subspace of R^q
Nul A is a subspace of R^m because solutions of Ax = 0 must have m entries
The set Nul A of all solutions of the homogenous equation Ax=0
Null Space of A
Forms of basis for the column space of A
Pivot columns of matrix A
The null space of an m x n matrix A is the subspace of
Rⁿ
The set of all solutions of a system Ax=0 of m homogenous linear equations in n unknown is a subspace of
Rⁿ
How would you construct a 3x4 matrix with rank 1
Since rank simply means pivot columns, create a 3x4 matrix with only one pivot column
Suppose columns 1,3,4,5, and 7 of a matrix A are linearly independent (but are not necessarily pivots) and the rank is 5. Explain why the five columns mentioned must be a basis for the column space of A
Since the rank is 5 that means there are 5 pivot columns. Since the 5 listed columns are linearly independent they must be the 5 pivot columns. Since the pivot columns of A for the basis of the columns space of A the 5 columns must for a basis for the column space of A
det A = ai1Ci1 + ai2Ci2 + ... + ainCin
The cofactor expansion across the ith row
det A = a1jC1j + a2jC2j + ... + anjCnj
The cofactor expansion down the nth column
If A is a triangular matrix, the set A is the product of the entires on
The main diagonal of A
The vector w is in the subspace generated by v₁ and v₂ if and only if
The vector equation x₁v₁ + x₂v₂ = w is consistant
Why is the set {v₁,...,v₅} in Rⁿ linearly dependent if dim Span {v₁,...v₅} = 4
There are 4 pivot columns since dimColA = 4 which means dimNulA = 1 not zero, hence there is a nullspace and the set is linearly dependent.
(True/False) A row replacement operation does not affect the determinate of a matrix
True
(True/False) Given vectors V1,...,Vp in Rⁿ the set of all linear combinations of these vectors is in subspace Rⁿ
True
(True/False) If the columns of A are linearly dependent, then det A = 0
True
(True/False) If two row interchanges are made in succession, then the new determinant equals the old determinant
True
(True/False) If v₁,....,vp are in Rⁿ then the Span{v1,....,vp} is the same as the column space of the matrix [v1....vp]
True
(True/False) Row operations do not affect linear dependence relations among the columns of a matrix
True
(True/False) The columns of an invertible n x n matrix form a basis for Rⁿ
True
(True/False) The determinant of A is the product of the pivots in any echelon form U of A, multiplex by (-1)^r, where r is the number of row interchanges made during row reduction from A to U
True
If A is an n x n matrix, then the transpose of the determinate of A is equal to...
det A
Let A be a square matrix, If a multiple of one row of A is added to another row to produce a matrix B, then set B equals
det A
How do you know if a vector p is in Nul A
if Ap = 0 it is in Nul A
Let A be a square matrix, if one row of A is multiplied by k to produce a matrix B, then set B equals
k * det A
