Linear Algebra MC Canvas

What are the three conditions for a matrix to be in a subspace?

1.Must contain the zero vector 2.Must be closed under vector addition 3.Must be closed under scaler multiplication

Suppose that A is a 4-by-8 matrix. What is the largest possible value of rank(A)?

4 The rank of a matrix A is the number of nonzero rows in any matrix B in row echelon form that is row equivalent to A. There can be at most 4 such rows.

Let A be a 7×9 matrix and suppose that Ax=b has a solution for all bb in R7R7. Determine the rank of A.

7 According to Theorem 1 in 2.2.8 ("Matrix, Rank and Consistency"), the system Ax=b has a solution for all b if and only if the rank of A is equal to the number of rows.

Suppose A is an invertible matrix and B is an inverse of A. Select the True statements.

A is a square matrix B is the only inverse of A If A has m rows, then for any choice of b in Rm, the linear system Ax = b has exactly one solution AB = BA

How can we geometrically represent the parametric equations x=2t y=−t+1 and z=t?

A line in R3.

In order to compute the matrix product AB, what must be true about the sizes of A and B?

A must have as many columns as B has rows.

We have a system of four linear equations in four variables. We can think about the graph of each equation as a 3-dimensional volume in R4. Which of the following could geometrically represent the solutions to this system?

A point in R4 A line in R4 A plane in R4 A three-dimensional volume in R4

Which of the following sets of vectors spans R2? (Select all that apply)

Any set containing two non-parallel vectors span R2. On the other hand, a set containing only parallel vectors spans only a line.

In some cases, a matrix may be row reduced to more than one matrix in reduced row echelon form, using different sequences of row operations.

False Using Theorem 2 from the notes, every matrix is row equivalent to a unique matrix in reduced row echelon form (RREF).

Let y be a nonzero vector, if Au=y and Av=y then u+v is a solution to the matrix equation Ax=y.

False, u+vu+v would be a solution to Ax=2y

.We find for a square coefficient matrix A, the homogeneous matrix equation, Ax=0 has only the trivial solution x=0. This means that

Matrix A has an inverse.

When can S be a finite subspace

The only finite subspace is the sent containing only the zero vector

Suppose a 4 x 5 matrix A has column vectors v1,v2,v3,v4,v5 If the rank of matrix A is 4, what can you say (select all that apply)?

The vectors v1,v2,v3,v4,v5 span R4 The Ax = b has a solution for all b. The rank of A is equal to the number of rows, therefore, by Theorem 2 from "Matrix Rank and Span", the system Ax=b has a solution for all b in R4 and the columns of A span R4.

If Ax=0\ is a homogeneous system of linear equations and A is a 3x3 matrix, a possible number of solutions to the system is

exactly one infinitely many

If A is an invertible matrix, what else must be true?

if AB = C then B = A^-1C 5A is invertible A^T is invertible A^2 is invertible.

system of three linear equations (in three variables)

no solution exactly one solution infinitely many solutions

A set of three planes may

not intersect intersect in one point intersect in a line intersect in a plane