3.6 Practice Problems

  1. What is \(24/3 + 5^2 - (8 -4)\)?
  2. What is \(\sum_{i = 1}^5 (i*3)\)?
  3. Take the derivative of \(f(x) =v(4x^2 + 6)^2\) with respect to \(x\).
  4. Take the derivative of \(f(x) = e^{2x + 3}\) with respect to \(x\).
  5. Take the derivative of \(f(x) = log (x + 3)^2\) with respect to \(x\).

Given \(X\) is an \(n\) x \(k\) matrix,

  1. \((X^{T}X)^{-1}X^{T}X\) can be simplified to?
  2. \(((X^{T}X)^{-1}X^{T})^{T} =\) ?
  3. If \(\nu\) is a constant, how does \((X^{T}X)^{-1}X^{T} \nu X(X^{T}X)^{-1}\) simplify?
  4. If a matrix \(P\) is idempotent, \(PP =\) ?

3.6.1 Practice Problem Solutions

  1. What is \(24/3 + 5^2 - (8 -4)\)?
24/3 + 5^2 - (8 -4)
[1] 29
  1. What is \(\sum_{i = 1}^5 (i*3)\)?
    • By hand: \(1 \times 3 + 2 \times 3 + 3 \times 3 + 4 \times 3 + 5 \times 3\)
## sol 1
1*3 + 2*3 + 3*3 + 4*3 + 5*3
[1] 45
## sol 2
i <- 1:5
sum(i*3)
[1] 45
  1. Take the derivative of \(f(x) =v(4x^2 + 6)^2\) with respect to \(x\).
    • We can treat \(v\) as a number.

\[\begin{align*} f'(x) &= 2* v(4x^2 + 6) * 8x\\ &= 16vx(4x^2 + 6) \end{align*}\]

  1. Take the derivative of \(f(x) = e^{2x + 3}\) with respect to \(x\).

\[\begin{align*} f'(x) &= 2* e^{2x + 3}\\ &= 2e^{2x + 3} \end{align*}\]

  1. Take the derivative of \(f(x) = log (x + 3)^2\) with respect to \(x\).
    • Note we can re-write this as \(2 * log (x + 3)\).

\[\begin{align*} f'(x) &= 2 * \frac{1}{(x + 3)} * 1\\ &= \frac{2}{(x + 3)} \end{align*}\] If we didn’t take that simplifying step, we can still solve:

\[\begin{align*} f'(x) &= \frac{1}{(x + 3)^2} * 2 * (x + 3) *1\\ &= \frac{2}{(x + 3)} \end{align*}\]

Given \(X\) is an \(n\) x \(k\) matrix,

  1. \((X^{T}X)^{-1}X^{T}X\) can be simplified to?
    • \(I_k\) the identity matrix
  2. \(((X^{T}X)^{-1}X^{T})^{T} =\) ?
    • Recall our rule \((AB)^T = B^TA^T\)
    • \(X(X^TX)^{-1}\)
  3. If \(\nu\) is a constant, how does \((X^{T}X)^{-1}X^{T} \nu X(X^{T}X)^{-1}\) simplify?
    • We can pull it out front.

\[\begin{align*} &= \nu(X^{T}X)^{-1}X^{T}X(X^{T}X)^{-1} \\ &= \nu (X^{T}X)^{-1} \end{align*}\]

  1. If a matrix \(P\) is idempotent, \(PP =\) ?
    • \(P\) from section 3.5.2