Hint for homework 1: Q7

For this question we are interested in simplifying an expression into matrix/vector notation. In order to do this it may be first helpful to think about how we went the other direction: matrix/vector notation to expanded notation.

Recall that a dot product between two vectors \(\mathbf{x}\) and \(\mathbf{y}\), can be written explicitly from its definition as:

\[ \mathbf{x}^T\mathbf{y} = \sum_{i=1}^n x_iy_i \]

Therefore if we see something like the summation on the right in an expression, we can replace it with the more compact dot product notation.

If we have the expression \(\mathbf{A} \mathbf{x}\), where \(\mathbf{A}\) is a matrix and \(\mathbf{x}\) is a vector, we know that the result of this multiplication will be a vector. Let’s call this vector \(\mathbf{c}\), so that \(\mathbf{A}\mathbf{x}=\mathbf{c}\).

We know from the definition of matrix-multiplication that each element of \(\mathbf{c}\) can be written as the following summation:

\[ c_i = \sum_{j=1}^n A_{ij}x_j \]

Therefore, if we saw such a summation in an expression, we could temporarily replace it with \(c_i\), knowing that we defined \(\mathbf{c}\) as \(\mathbf{c}=\mathbf{A}\mathbf{x}\). Try doing this as the first step in the homework, then try repeating this idea until you have something that you can write compactly in matrix/vector notation.

When writing you final answer make sure to write the final expression in terms of the original variables \((\mathbf{A}, \mathbf{x}, \mathbf{b})\). E.g. if you substituted \(\mathbf{c}\) for \(\mathbf{A}\mathbf{x}\), make sure to substitute it back in the answer.