Are U V and W linearly independent?

Since u,v,w are linearly independent, {a−b−c=0−a+b−c=0−a−b+c=0.

Will u v/v W and U W be linearly independent as well justify your answer?

Suppose {v1,v2,v3} is linearly dependent. Then by definition, at least one of the vectors v1, v2, or v3 is a linear combination of the other two. Without loss of generality, let’s relabel the vectors so that v3 is a linear combination of v1 and v2. That is, v3 = c1v1 +c2v2 for some c1,c2 ∈ R.

How do you show that two vectors are linearly independent?

We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.

Is vector R2 a subspace of R3?

If U is a vector space, using the same definition of addition and scalar multiplication as V, then U is called a subspace of V. However, R2 is not a subspace of R3, since the elements of R2 have exactly two entries, while the elements of R3 have exactly three entries. That is to say, R2 is not a subset of R3.

What can you say about the linear span of the empty set?

The span of the empty set is the set containing just the zero vector. Theorem: If S is any subset of V , the span of S is the smallest linear subspace of V containing S.

Are all orthogonal sets linearly independent?

A nonempty subset of nonzero vectors in Rn is called an orthogonal set if every pair of distinct vectors in the set is orthogonal. Orthogonal sets are automatically linearly independent. Theorem Any orthogonal set of vectors is linearly independent.

Why is P2 a subspace of P3?

Since every polynomial of degree up to 2 is also a polynomial of degree up to 3, P2 is a subset of P3. And we already know that P2 is a vector space, so it is a subspace of P3.

How do you prove that a basis is V?

Let V be a vector space of dimension n. (1) If S is a linearly independent subset of V and S contains n (distinct) vectors, then S is a basis of V . (2) If S is a subset of V such that V = span (S ) and S contains n vectors, then S is a basis of V . Proof.