notes5-4

# notes5-4 - ◦ deﬁnes a linear transformation which...

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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 .

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EXAMPLE. Remember from long ago (Section 1.8) that rotation about the origin through an angle of 60
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Unformatted text preview: ◦ deﬁnes a linear transformation which we’ll 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. Diﬀerentiation 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 diﬀerent bases! What does all this say about matrices that are diagonalizable? HOMEWORK: SECTION 5.4, #2, 4, 6, 8, 12, 14, 18...
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notes5-4 - ◦ deﬁnes a linear transformation which...

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