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Exercises $9-12$ display a matrix $A$ and an echelon form of $A$ . Find bases for $\operatorname{Col} A$ and $\mathrm{Nul} A,$ and then state the dimensions of these subspaces.$$A=\left[\begin{array}{rrrr}{1} & {-3} & {2} & {-4} \\ {-3} & {9} & {-1} & {5} \\ {2} & {-6} & {4} & {-3} \\ {-4} & {12} & {2} & {7}\end{array}\right] \sim\left[\begin{array}{rrrr}{1} & {-3} & {2} & {-4} \\ {0} & {0} & {5} & {-7} \\ {0} & {0} & {0} & {5} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

$\left[\begin{array}{c}{1} \\ {-3} \\ {2} \\ {-4}\end{array}\right],\left[\begin{array}{c}{2} \\ {-1} \\ {4} \\ {2}\end{array}\right],\left[\begin{array}{c}{-4} \\ {5} \\ {-3} \\ {7}\end{array}\right]$3$\left[\begin{array}{l}{3} \\ {1} \\ {0} \\ {0}\end{array}\right]$1

Algebra

Chapter 2

Matrix Algebra

Section 9

Dimension and Rank

Introduction to Matrices

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

03:25

Exercises $9-12$ display a…

03:13

07:02

Exercises $23-26$ display …

07:04

08:16

Find an orthogonal basis f…

01:57

Find bases for the null sp…

16:00

06:08

In Exercises 11 and $12,$ …

06:21

11:56

In Exercises 37 and 38, co…

again in this question, You want to find the color space and no spice all this matrix right here. So to find the column space, all we need to do is to find a pivot follows off this matrix right here at this difficult. The pivot columns are the same pivot columns as those in its road echelon formed, right? Yeah. So we have about it. Havoc, homes being these three forms. So therefore, our column Ah, calmness face is simply columns face for a He's a span. All of these four metres. 154 minds too native four muted mine medicated wine five and finally 387 and six To find a null space off A you take its right echelon form and you want to Seoul. So if you said this is going to be you want to solve b X is it itches. So in parametric form, you want to solve 002000 minus 4100 three months. 20030 minus five is there and also rose. So then you wanna solve x one x two x three Export and x So you know in what this line tells you is x one plus two x two minus four extra plus three Export plus three x five is equal to zero. Just sold for X one you put minus two to decide. Plus fallin decide my street and my street. So this is going to be negative two x two plus by four x three minus three x four minus three x so x two There is no people calling for extra extra is just simply x two x three is going to be just simply two x four to export. Would you get from which he gets from this line right here. Now explore also had people call him. So export is just X for and X five we get from this one, which is negative X five music to zero x five is just is there So our mole space No space over a moment. Okay, Is the span off these actors? Minus 2000 And this is one, um full. 0000 Negative. 30 210 If and finally native 3000

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In mathematics, the absolute value or modulus |x| of a real number x is its …

Exercises $9-12$ display a matrix $A$ and an echelon form of $A$ . Find base…

Exercises $23-26$ display a matrix $A$ and an echelon form of $A .$ Find a b…

Find an orthogonal basis for the column space of each matrix in Exercises $9…

Find bases for the null spaces of the matrices given in Exercises 9 and $10 …

In Exercises 11 and $12,$ find the dimension of the subspace spanned by the …

In Exercises 37 and 38, construct bases for the column space and the null sp…

02:01

Determine by inspection whether the vectors are linearly independent. Justif…

01:33

Find the determinants in Exercises $15-20,$ where$$\left|\begin{arra…

03:52

Consider the production model $\mathbf{x}=C \mathbf{x}+\mathbf{d}$ for an ec…

01:09

Compute the determinants in Exercises $9-14$ by cofactor expansions. At each…

01:43

In Exercises $3-8,$ find the $3 \times 3$ matrices that produce the describe…

07:38

With $A$ and $\mathbf{p}$ as in Exercise $7,$ determine if $\mathbf{p}$ is i…

01:24

Determine which sets in Exercises $15-20$ are bases for $\mathbb{R}^{2}$ or …

01:41

Compute the determinants in Exercises $1-8$ using a cofactor expansion acros…

05:13

Exercises $22-26$ provide a glimpse of some widely used matrix factorization…

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