Subspaces and bases
This presentation is the property of its rightful owner.
Sponsored Links
1 / 8

SUBSPACES and BASES PowerPoint PPT Presentation


  • 45 Views
  • Uploaded on
  • Presentation posted in: General

SUBSPACES and BASES. 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

Download Presentation

SUBSPACES and BASES

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Subspaces and bases

SUBSPACES and BASES

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


Subspaces and bases

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


Subspaces and bases

(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


Subspaces and bases

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

and

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


Subspaces and bases

(Visual representation of a linear transformation)

We give the following definitions:


Subspaces and bases

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:


Subspaces and bases

(Span)

(basis)

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


Subspaces and bases

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.


  • Login