Define the dimension of a finite-dimensional vector space \(V\).

What is \(\dim\mathbb R^4\)?

What is the dimension of the zero subspace \(\{\mathbf0\}\)?

What is the dimension of \(L=\{t(3,-1):t\in\mathbb R\}\)?

Find the dimension of \(W=\{(x,y,0):x,y\in\mathbb R\}\).

Find the dimension of \(\operatorname{span}\{(1,2,3)\}\).

What is the dimension of \(P_2\), the polynomials of degree at most \(2\)?

Find the dimension of \(\operatorname{span}\{(1,0),(0,1),(1,1)\}\) in \(\mathbb R^2\).

Find the dimension of \(W=\{(x,y,z):x+y+z=0\}\).

Find the dimension of \(\{p\in P_3:p(0)=0\}\).

Explain why a plane through the origin in \(\mathbb R^3\) has dimension \(2\).

A particle constrained to move on a line through the origin in space has how many positional degrees of freedom?

Find the dimension of all real \(2\times2\) diagonal matrices.

Find the dimension of all real \(2\times2\) symmetric matrices.

For which \(a\) does \(\operatorname{span}\{(1,0,a),(0,1,a),(1,1,2a)\}\) have dimension \(2\)?

For which \(t\) does \(\operatorname{span}\{(1,0,0),(0,1,0),(1,1,t)\}\) have dimension \(3\)?

For which \(b\) does \(\operatorname{span}\{(1,b),(b,1)\}\) have dimension \(1\)?

A student counts three vectors in a spanning list and says the dimension is \(3\). Give a correction using \((1,0),(0,1),(1,1)\).

Explain why a subspace of \(\mathbb R^3\) cannot have dimension \(4\).

Prove that if \(U\subseteq V\) are finite-dimensional vector spaces, then \(\dim U\le\dim V\).