Algebra lineal y algunas de sus aplicaciones by L.I. GOLOVINA

By L.I. GOLOVINA

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In particular, the (i, j) entry of (λA)T is λa ji , which is also the (i, j) entry of λAT . The next property tells you what happens when you take the transpose of a product of matrices: r (AB)T = B T AT . 22 Matrices and vectors This can be stated as: The transpose of the product of two matrices is the product of the transposes in the reverse order. Showing that this is true is slightly more complicated, since it involves matrix multiplication. It is more important to understand why the product of the transposes must be in the reverse order: the following activity explores this.

For example, if t = 2, we get the position vector of the point (2, 4) on the line, and if t = −1, we obtain the point (−1, −2). As the parameter t runs through all real numbers, this vector equation gives the position vectors of all the points on the line. 9 The line y = 2x + 1. The vector shown is v = (1, 2)T y x (0, 0) we can retrieve the Cartesian equation using the fact that the two vectors are equal if and only if their components are equal. This gives us the two equations x = t and y = 2t.

In particular, the (i, j) entry of (λA)T is λa ji , which is also the (i, j) entry of λAT . The next property tells you what happens when you take the transpose of a product of matrices: r (AB)T = B T AT . 22 Matrices and vectors This can be stated as: The transpose of the product of two matrices is the product of the transposes in the reverse order. Showing that this is true is slightly more complicated, since it involves matrix multiplication. It is more important to understand why the product of the transposes must be in the reverse order: the following activity explores this.

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