Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W = R". The goal is to show that the subspace W is actually all of R". In general, when are two subspaces known to be the same space? O A. Two subspaces are the same when they are spanned by the same vectors. O B. Two subspaces are the same when they have the same dimension. Oc. Two subspaces are the same when one subspace is a subset of the other subspace. O D. Two subspaces are the same when they are subsets of the same space. To start, assume that W is a subspace of R" spanned by n orthogonal nonzero vectors. Call these vectors v,, V2, V3. ., Vo: By assumption, W= Span (v,, V2, V3, ..., V. Which of the following statements follows immediately from the assumption that the vectors v,, v2, V3, .., V, are orthogonal and nonzero? O A. At least one element of (v1, V2, V3, .., Vn) is zero. O B. R" = Span{V1, V2. V3. .. Vn} Oc. (V1, V2. V3, ..., Vn) is linearly dependent. O D. (V1, V2. V3, ..., V is linearly independent. How does this show that W = R"? O A. The vectors V. V2. V3. ,Vn are in both W and R". O B. The column vectors v,, V2, V3, .., V, span R" if and only if (v1, V2, V3, ., Vn is linearly dependent. O C. The column vectors v1, V2, V3. ., Vn span R" if and only if {V1, V2, V3, ..., Vn is linearly independent. O D. w= (V1, V2, V31 ..., Vn} = R".

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.