Week 5 Checkpoint

$\{(1,2),\ (2,4)\}$

$\{(1,0),\ (0,1)\}$

$\{(3,1),\ (6,2)\}$

$\{(1,1),\ (-2,-2)\}$

Rank $3$, column-space dimension $3$

Rank $1$, column-space dimension $1$

Rank $0$, column-space dimension $0$

Rank $2$, column-space dimension $1$

Subspace of $\mathbb{R}^m$; it means $A$ is invertible

Subspace of $\mathbb{R}^n$; it means the columns of $A$ are linearly dependent

Subspace of $\mathbb{R}^m$; it means the rows are orthonormal

Subspace of $\mathbb{R}^n$; it means $A$ has full column rank

$Av=(3,3)$, so $\lambda=3$

$Av=(2,2)$, so $\lambda=2$

$Av=(1,1)$, so $\lambda=1$

$v$ is not an eigenvector of $A$

The two features are uncorrelated

When feature 1 is above its mean, feature 2 tends to be below its mean

Feature 1 has larger variance than feature 2

The two features are linearly independent as vectors

$1/3\approx 33\%$

$8/10 = 80\%$

$8/9\approx 89\%$

$1/8 = 12.5\%$

minimizes the variance of the projected data

is the eigenvector of the covariance matrix with the largest eigenvalue

is always aligned with the first coordinate axis

corresponds to the smallest eigenvalue of the covariance matrix