Let A=( 1 -2 2 -1 -2 4 -3 5 -1 2 0 7 ). (a)

step 1 Okay, so here we have that matrix A is the matrix here. One, two, three, one, two, four. So look

Let A=( 1 -2 2 -1 -2 4 -3 ...

Let $A=\left(\begin{array}{rrrr}1 & -2 & 2 & -1 \\ -2 & 4 & -3 & 5 \\ -1 & 2 & 0 & 7\end{array}\right)$. (a) Find a basis for coimg $A$. (b) Use Theorem 4.49 to find a basis of $\operatorname{img} A$. (c) Write each column of $A$ as a linear combination of the basis vectors you found in part $(b)$. $\uparrow$ An alternative is to orthogonally project the general solution onto the coimage. The result is the same.

Let $A=\left(\begin{array}{rrrr}1 & -2 & 2 & -1 \\ -2 & 4 & -3 & 5 \\ -1 & 2 & 0 & 7\end{array}\right)$.
(a) Find a basis for coimg $A$. (b) Use Theorem 4.49 to find a basis of $\operatorname{img} A$. (c) Write each column of $A$ as a linear combination of the basis vectors you found in part $(b)$.
$\uparrow$ An alternative is to orthogonally project the general solution onto the coimage. The result is the same.

Key Concepts

Coimage

The coimage of a linear transformation is the quotient of its domain by the kernel. It represents the part of the domain where the transformation acts injectively, and—it turns out—is isomorphic to the image. This concept helps in understanding how redundancy from the kernel is removed to obtain a one?to?one correspondence with the image.

Image

The image of a linear transformation is the set of all vectors in the codomain that can be expressed as the transformation of some vector in the domain. In the context of matrices, this is also known as the column space. It is a subspace of the codomain and its basis can be found by identifying the minimal set of vectors that span all outputs.

Basis

A basis of a vector space is a set of linearly independent vectors that span the entire space. It serves as a minimal generating set that allows every element of the space to be uniquely represented as a linear combination of the basis vectors. Basis selection is crucial for simplifying many problems in linear algebra.

Linear Combination

A linear combination is an expression made up of sums of scalar multiples of vectors. In the context of vector spaces, representing vectors as linear combinations of a given basis allows for a clear and structured understanding of the space and facilitates solving linear systems and understanding transformations.

Orthogonal Projection

Orthogonal projection is the operation of projecting a vector onto a subspace along directions orthogonal to that subspace. This is particularly useful in decomposing vectors relative to the basis of a subspace and is an alternative method in finding representations and simplifying computations in linear algebra.

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