State the definition of a linearly independent list \(\mathbf v_1,\ldots,\mathbf v_k\).

What does it mean for a list of vectors to be linearly dependent?

Explain why any list containing \(\mathbf0\) is linearly dependent.

Are \((1,0)\) and \((0,1)\) linearly independent in \(\mathbb R^2\)?

Test whether \((1,2)\) and \((3,6)\) are linearly independent.

Test whether \((2,1)\) and \((1,-1)\) are linearly independent.

Explain why three vectors in \(\mathbb R^2\) must be linearly dependent.

Test whether \((1,0,0),(0,1,0),(1,1,0)\) are linearly independent.

Test whether \((1,0,1),(0,1,1),(1,1,0)\) are linearly independent.

Test whether the polynomials \(1+x\), \(1-x\), and \(2\) are linearly independent in \(P_1\).

Use a determinant to test whether \((1,2)\) and \((3,5)\) are independent in \(\mathbb R^2\).

A pair of non-zero displacement directions in a plane are not scalar multiples. Explain why they are independent.

Find all \(k\) for which \((1,k)\) and \((2,4)\) are linearly dependent.

Find all \(a\) for which \((1,0,a),(0,1,a),(1,1,2a)\) are linearly independent.

For which \(t\) are \((1,0,1),(0,1,1),(1,1,t)\) linearly independent?

For which \(a\) are \((1,a)\) and \((a,1)\) linearly dependent?

For which \(p\) are the polynomials \(1+x\), \(1+px\) independent in \(P_1\)?

A student checks that no two vectors in a list of three are scalar multiples and concludes the list is independent. Give a counterexample.

Prove that any subset of a linearly independent list is linearly independent.

Prove that if one vector in a finite list is a linear combination of the others, then the list is linearly dependent.