Exercise 590Linear algebra › Vector spaces and subspacesHigh schoolAdded 3 Nov 2026

Recognizing bases

#basis#dimension#canonical basis

Statement

Which of the following families are bases of $\mathbb{R}^3$ ?

(a) $(1,0,0), (0,1,0), (0,0,1)$ ;
(b) $(1,1,0), (0,1,1), (1,0,1)$ ;
(c) $(1,2,3), (2,4,6), (0,1,0)$ ;
(d) $(1,0,0), (0,1,0)$ .

Hint

A basis of $\mathbb{R}^3$ must have exactly $3$ linearly independent vectors.

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(a) The canonical basis: independent and has $3$ vectors. Basis.

(b) The determinant is $\begin{vmatrix}1&0&1\\1&1&0\\0&1&1\end{vmatrix} = 1(1-0)-0+1(1-0) = 2\neq 0$ . Independent, $3$ vectors. Basis.

(c) The second vector is twice the first, so the family is dependent. Not a basis.

(d) Only $2$ vectors, not enough to span $\mathbb{R}^3$ . Not a basis.

Conclusion. (a) and (b) are bases; (c) and (d) are not.