notes5-4 - denes a linear transformation which well call T...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
SECTION 5.4 LINEAR TRANSFORMATIONS AND MATRICES AGAIN Remember that if A is a p × q matrix, then T ( x ) = A x defines a linear transformation from R q to R p . But there are linear transformations, like rotations, that are not given by matrix multiplication, and what about linear transformations from an abstract vector space V to another abstract vector space W ? Suppose that V is a vector space with basis B having q vectors, that W is a vector space with basis C having p vectors, and finally that T is a linear transformation from V to W . Pause a moment and consider what this means. Remember that coordinate mappings connect V with R q and W with R p . Let’s use these coordinate mappings to analyze T . The matrix M consisting of the C -coordinates of the T -images of the b ’s is called the matrix for T relative to the bases B and C . It has the property that [ T ( x )] C = M [ x ] B .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
EXAMPLE. Remember from long ago (Section 1.8) that rotation about the origin through an angle of 60
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: denes a linear transformation which well call T . We now know that there is a matrix M so that T ( x ) = M x , where we use the standard basis on both sides. What is M ? EXAMPLE. Dierentiation is a linear transformation from P 3 to P 2 . What is its matrix relative to the standard bases? It often happens that the linear transformation goes from a vector space V to the same vector space V . We then need just one basis B and we get the matrix for T relative to B , or just the B-matrix for T . What happens to the matrix when we change bases? The matrices similar to a matrix A are exactly those matrix representations of the linear transformation T ( x ) = A x relative to dierent bases! What does all this say about matrices that are diagonalizable? HOMEWORK: SECTION 5.4, #2, 4, 6, 8, 12, 14, 18...
View Full Document

Page1 / 3

notes5-4 - denes a linear transformation which well call T...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online