matrix of a linear transformation

# matrix of a linear transformation - The matrix of a linear...

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Unformatted text preview: The matrix of a linear transformation In the case where V and W are finite-dimensional vector spaces over the field F , we can identify V with F n and W with F m ( n and m being the dimensions of V and W over F , respectively). To do this, we need to fix an ordered basis B of V and one (say C ) of W . The identifications (isomorphisms, really) are then ~v 7→ [ ~v ] B and ~w 7→ [ ~w ] C . As we have seen, this kind of thing depends on all the data; the field, the vector spaces, and crucially the chosen bases. Now suppose that T is a linear transformation from V to W and we have fixed an ordered basis B = ( ~v 1 ,...,~v n ) of V and an ordered basis C = ( ~w 1 ,..., ~w m ) for W . (We are assuming that m and n are finite here.) DEFINITION (The matrix of the linear transformation T with re- spect to the ordered bases B and C ) For each 1 ≤ j ≤ n , the j th column of the matrix which we will denote C [ T ] B is the m × n matrix such that its j th column (for j = 1 ,...,n ) will be [ T~v j ] C . That is, we evaluate the j th vector in the basis B and express it in terms of the basis C . (Others use different notation.) In the important special case where V = W and B = C , we will just write [ T ] B instead of B [ T ] B except occasionally for emphasis. A trivial example is when T = 0 V,W . Whatever ordered bases B and C we choose, we get C [ T ] B = 0 m,n . Also, if V = W and T = αI V , then for any B , [ T ] B = αI n . Another simple observation is that if T = T A for an m × n matrix A , V = F n , W = F m and we choose B and C as the standard bases for each of V and W , then C [ T A ] B is just — aw, you guessed — A itself. (If we vary the bases, it probably won’t be, though.) Consider the projection Proj ‘ and reflection T ‘ (where ‘ is a line through the origin in the real space R 2 ). Instead of choosing the standard basis here, let’s suppose B = C is a basis consisting of a vector ~v 1 on ‘ and a vector ~v 2 on ‘ ⊥ . Then Proj ‘ ~v 1 = ~v 1 = 1 ~v 1 + 0 ~v 2 and Proj ‘ ~v 2 = ~ 0. Also T ‘ ~v 1 = ~v 1 and T ‘ ~v 2 =- ~v 2 . We get particularly simple matrices this way; [ Proj ‘ ] B = 1 0 0 0 ¶ and [ T ‘ ] B = 1- 1 ¶ . If we want the matrices with respect to the standard basis ( ~e 1 ,~e 2 ), it still helps to refer to this nonstandard basis; well, it’s nonstandard unless ‘ is the x-axis. Let’s do this in the specific case where ‘ is defined by y = 2 3 x . Let ~v 1 = 3 2 ¶ on ‘ . (I’m tempted to divide this by √ 13 for some reason, but let’s forgo that for now.) Let ~v 2 =- 2 3 ¶ on ‘ ⊥ . Now ~e 1 = 3 13 ~v 1 +- 2 13 ~v 2 and ~e 2 = 2 13 ~v 1 + 3 13 ~v 2 . So Proj ‘ ~e 1 = 3 13 ~v 1 = 1 13 9 6 ¶ , Proj ‘ ~e 2 = 1 13 6 4 ¶ ....
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matrix of a linear transformation - The matrix of a linear...

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