Homework Ch 3.6 Quizlet

7. Let A be a 6 × n matrix of rank r and let b be a vector in R^6. For each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. d. n=5, r=4

Either no solutions or infinitely many solutions

rowspace of a matrix A(mxn) columnspace of a matrix A(mxn)

rowspace: The span of all row vectors of A columnspace: The span of all column vectors of A

Range/ Image of matrix A

the column space of A= {Ac | c e R} Note the row space of A= {A^Ty | y e R^n}, the image of A^T is not necessarily accurate

What is nullity(A)?

the dimension of the null space of A (dim(N(A)) Remember null space: Ax=0

2. In each of the following, determine the dimension of the subspace of R3 spanned by the given vectors. a) ⎧ 1 ⎫⎧2 ⎫⎧-3⎫ ⎪-2⎪⎪-2⎪⎪ 3⎪ ⎩ 2⎭,⎩4⎭,⎩ 6⎭

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2. In each of the following, determine the dimension of the subspace of R3 spanned by the given vectors. b) ⎧ 1 ⎫⎧ 1 ⎫⎧ 2⎫ ⎪ 1 ⎪⎪ 2⎪⎪ 3⎪ ⎩ 1 ⎭,⎩3⎭,⎩ 1⎭

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7. Let A be a 6 × n matrix of rank r and let b be a vector in R^6. For each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. a. n=7, r=5

Either infinite or no solutions

7. Let A be a 6 × n matrix of rank r and let b be a vector in R^6. For each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. c. n=5, r=5

Either no solution or a unique solution

1. For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space. b. ⎧ -3 1 3 4 ⎫ ⎪ 1 2 -1 -2 ⎪ ⎩-3 8 4 2 ⎭

Find reduced r/e form of the matrix ⎧ 1 0 0 -10/7⎫ ⎪ 0 1 0 -2/7⎪ ⎩ 0 0 1 0 ⎭

Fundamental Theorem of Linear Algebra

rank(A) + nullity(A) = n this is, by definition, saying the number of leading variables + the number of free variables = total number of rows in the matrix

What can we determine about the dimension of the row-space of A and the dimension of the column space of A

They are the same rank(A)=rank(A^T)

Method of finding nullity(A)

Use row operation to transform a matrix into a row-echelon matrix. The number of free variables is the nullity(A)

Method of finding rank(A)

Use row-operation to transform A into a row-echelon matrix The number of leading entries is the rank(A)

A linear system Ax = b is consistent if and only if

b is in the column space of A

What does a "consistent" linear system mean

It has at least one solution. The system Ax=b will be consistent if/f b is an element of the column space of A

What is rank(A)

The Rank of matrix A, denoted by Rank(A) is the dimension of the row space. It is the biggest number of rows that are linearly independent (so we can have a huge matrix, but its rank may be very small)

2. In each of the following, determine the dimension of the subspace of R3 spanned by the given vectors. c) ⎧ 1 ⎫⎧-2 ⎫⎧ 3⎫⎧ 2⎫ ⎪-1⎪ ⎪ 2 ⎪⎪-2⎪⎪-1⎪ ⎩ 2⎭,⎩-4⎭,⎩5⎭,⎩ 3⎭

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1. For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space. c. ⎧ 1 3 -2 1 ⎫ ⎪ 2 1 3 2 ⎪ ⎩ 3 4 5 6 ⎭

Find reduced r/e form of the matrix ⎧ 1 0 0 -13/20⎫ ⎪ 0 1 0 21/20 ⎪ ⎩ 0 0 1 3/4 ⎭ Possible Basis of row space: {(1,0,0,−13/20​), (0,1,0,21/20​) ,(0,0,1,3/4​)​} Possible basis for column space: {(1,2,3^T, (3,1,4)^T, (−2,3,5)^T​}

1. For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space. a. ⎧ 1 3 2 ⎫ ⎪ 2 1 4 ⎪ ⎩4 7 8 ⎭

Find reduced r/e form of the matrix (R2-2R1, R3-4R1, R3-R2, (-5)R2, R1-3R2) to get ⎧ 1 0 2 ⎫ ⎪ 0 1 0 ⎪ ⎩ 0 0 0 ⎭

6. How many solutions will the linear system Ax = b have if b is in the column space of A and the column vectors of A are linearly dependent? Explain.

Infinitely many solutions

7. Let A be a 6 × n matrix of rank r and let b be a vector in R^6. For each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. b. n=7 r=6

Infinitely many solutions

a) Let A be an m×n matrix. The linear system Ax = b is consistent for every b ∈ R^m if and only if the column vectors of A... b)The system Ax = b has at most one solution for every b ∈ Rm if and only if the column vectors of A

a) span R^m b) are linearly independent

The system Ax=b is consistent if and only if

b is an element of the column space of A

An n × n matrix A is nonsingular if and only if the column vectors of A...

form a basis for Rn.