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SUBSPACES and BASES

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We saw last the notion of a subspace W of a vector space V. Recall:

Definition:

The examples we have seen so far originated from considering the span of the column vectors of a matrix A, or the solution set of the equation

We called the former

We studied how to find bases of both, and compute their respective dimensions.

What we did not stress at the time was “which vector spaces

Let A be an

On the other hand, a vector

(shown some time ago)

I highly recommend to you studying the very nice table on p. 204 of the textbook that lists 8 items of contrast between and for an

matrix A.

If a vector space is not obviously

life isn’t as easy, subspaces arise mostly in connection with .

Let’s introduce a precise Definition:

Given two vector spaces V and W, a linear transformation from V into W is a function

(the textbook says a rule that … we simply write)

and

The figure shown in the next slide will be useful in the future.

(Visual representation of a linear transformation)

We give the following definitions:

We define kernel of T(or null space of T) the set

The subset of W defined by

Is called the range of T, also the image of T.

It’s a simple exercise to show that the kernel of T is a subspace of V, and that the range of T is a subspace of W.

The notions of Span of a set of vectors and of basis are defined for any vector space as usual:

Here are the two definitions:

(Span)

(basis)

The following theorem (5 in the textbook, p. 210)

has two very interesting consequences. Here is the

Theorem.

Then

.

The proof is very easy, we’ll do it on the board.

Here are the two interesting consequences:

A basis of H is, equivalently, a minimal spanning set of H and/or

a maximal linearly independent subset of H.