Linear Algebra for Machine Learning and Data Science (12)

Vectors and Linear Transformations

Check your knowledge

Q1

If $\vec u=(3,6)$ and $\vec v=(5,2)$, compute $\vec{u}+\vec{v}$ and $\vec{u}-\vec{v}$. (Videos: Vectors and their properties, Sum and difference of vectors)

Addition: (8, 8)

Subtraction: (-2, 4)

Q2

What is the distance between vectors $\vec u$ and $\vec v$? (Video: Distance between vectors)

L1-distance: 6

L2-distance: $\sqrt{20}$

Q3

If $\vec u = (3,5)$ compute 6$\vec u$. (Video: Multiplying a vector by a scalar)

(18, 30)

Q4

If $\vec u=(1,2,3)$ and $\vec v=(0,3,5)$, compute the dot product $\vec u⋅ \vec v$.

21

Q5

If $\vec v=\left[\begin{array}{c}1\\2\\5\end{array}\right]$ and $M = \left[\begin{array}{ccc}4&5&9\\7&1&0\\4&2&1\end{array}\right]$, compute $M \vec v$. (Video: Multiplying a matrix by a vector)

$\left[\begin{array}{c}59\\9\\13\end{array}\right]$

Q6

If $T$ is a linear transformation defined as $T(0,1)=(1,2)$ and $T(1,0)=(4,1)$, find the matrix that represents this transformation. (Videos: Matrices as linear transformations, Linear transformations as matrices)

$\left[ \begin{array}{cc}4&1\\1&2 \end{array} \right]$

Q7

If $M = \left[ \begin{array}{cc}1&2\\3&5 \end{array} \right]$ and $N = \left[ \begin{array}{cc}4&0\\1&2 \end{array} \right]$, compute $M \cdot N$ and $N^{-1}$. (Videos: Matrix multiplication, The identity matrix, Matrix inverse, Which matrices have an inverse)

$M\cdot N=\left[ \begin{array}{cc}6&4\\17&10 \end{array} \right]$ and $N^{-1}= \left[ \begin{array}{cc}{1\over4}&0\\-{1\over8}&{1\over2} \end{array} \right]$

All the information here is based on the Linear Algebra for Machine Learning and Data Science | Coursera from DeepLearning.AI