Assume that A is row equivalent to B. Find bases for Nul A and Col A.

djeljenike
2021-09-13
Answered

Assume that A is row equivalent to B. Find bases for Nul A and Col A.

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BleabyinfibiaG

Answered 2021-09-14
Author has **118** answers

Since B is a row echelon form of A, we see that the first, third and fifth columns of A are its pivot columns. Thus a basis fo Col A is

To find a basis for Nul A, we find the general solution of Ax=0 in terms of the free variables. Since it is row equivalent to B we can simply get reduced row echelon form of B:

to get:

And a basis for Nul A is

Result: Basis for Col A is:

Basis for Nul A is

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For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a nonzero vector in Nul A.

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Find a nonzero vector in Nul A.

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Let $v}_{1},{v}_{2},\dots .,{v}_{k$ be vectors of Rn such that

$v={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\dots +{c}_{k}{v}_{k}={d}_{1}{v}_{1}+{d}_{2}{v}_{2}+\dots +{d}_{k}{v}_{k}$ .

for some scalars$c}_{1},{c}_{2},\dots .,{c}_{k},{d}_{1},{d}_{2},\dots .,{d}_{k$ .Prove that if $ci\ne dif{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}somei=1,2,\dots .,k$ ,

then$v}_{1},{v}_{2},\dots .,{v}_{k$ are linearly dependent.

for some scalars

then

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A linear transformation acts on ${R}^{2}$ by first doubling the x-coordinate and then rotating the plane by an angle of $\pi /2$ (${90}^{\circ}$) counter-clockwise. What is the corresponding matrix to this linear transformation?

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Find the equation of the line that satisfies the given conditions:

Goes through $(-\frac{1}{2},\frac{4}{5})$ and is perpendicular to the line $4x-5y=10$

Goes through $(-\frac{1}{2},\frac{4}{5})$ and is perpendicular to the line $4x-5y=10$

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Take, for instance, the following matrix:

$\left[\begin{array}{cc}12& 5\\ 5& -12\end{array}\right]$

How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

$\left[\begin{array}{cc}12& 5\\ 5& -12\end{array}\right]$

How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

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(a) Find the bases and dimension for the subspace