1.5. The Matrix of a Linear Transformation 5 Matrix multiplication and addition satisfy the following properties: • Multiplication is associative: A(BC) = (AB)C. • Distributive property: A(B + C) = AB + AC and (B + C)A = BA + CA. Matrix multiplication is generally not commutative. Also, the product of two nonzero matrices may be zero, for example 3 −3 5 −5 1 1 1 1 = 0 0 0 0 . The n×n matrix with ones on the main diagonal and zeroes elsewhere is denoted In and is called the identity matrix, for it serves as the multiplicative identity for n × n matrices. We use 0m×n to denote an m × n matrix of zeroes note that for any m × n matrix A, we have A + 0m×n = A. We will omit the subscripts on the identity and zero matrices when the size is clear from the context. 1.5. The Matrix of a Linear Transformation Given an m × n matrix A, we can define a linear transformation T : Fn → Fm by the rule T (x) = Ax. In fact, any linear transformation on finite dimensional vector spaces can be represented by such a matrix equation, relative to a choice of bases for the vector spaces. We now describe this in more detail and introduce some notation for the matrix of a linear transformation, relative to a choice of bases. Let V and W be finite-dimensional vector spaces over F. Let n = dim(V) and m = dim(W), and suppose T : V → W is a linear transformation. Suppose we have a basis B = {v1,..., vn} for V and a basis C = {w1,..., wm} for W. For any x ∈ V, we have x = n ∑ j=1 xjvj, and thus T (x) = ∑n i=1 xjT (vj). Hence [T (x)]C = ∑n j=1 xj[T (vj)]C. Let A be the m×n matrix which has [T (vj)]C in column j and note that [x]B = ⎛ ⎜ ⎜ ⎝ x1 x2 . . xn ⎞ ⎟ ⎟. ⎠ We then have [T (x)]C = A[x]B. We say the matrix A represents T relative to the bases B and C, and we write [T ]B,C = A. In the case where V = W and B = C, we write [T ]B = A. Most of the time, when we have V = Fn, W = Fm, and are using the standard bases for Fn and Fm, we identify the linear transformation T with the matrix A and simply write T (x) = Ax, with x regarded as a column vector of length n. The rank of a matrix may be defined in several equivalent ways. Let A be an m × n matrix, and let T be the linear transformation T (x) = Ax. We can then define the rank of A to be the rank of T . Alternatively, one can consider the subspace spanned by the n columns of A this is called the column space and its dimension is called the column rank of A. Clearly, the column rank equals the rank of T . The space spanned by the m rows of A is called the row space of A its dimension is called the row rank. When A is the coeﬃcient matrix for a system of m linear

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