Problem 5: Let A € M2×2(R). For u, v € R², define fa(u, v) = ut Av. 1 Here we are taking u and v to be 2 x 1 column vectors. Show that fA defines an inner product on R2 if and only if A = At, a11 > 0, α22 > 0, and det A > 0.
Problem 5: Let A € M2×2(R). For u, v € R², define fa(u, v) = ut Av. 1 Here we are taking u and v to be 2 x 1 column vectors. Show that fA defines an inner product on R2 if and only if A = At, a11 > 0, α22 > 0, and det A > 0.
Problem 5: Let A € M2×2(R). For u, v € R², define fa(u, v) = ut Av. 1 Here we are taking u and v to be 2 x 1 column vectors. Show that fA defines an inner product on R2 if and only if A = At, a11 > 0, α22 > 0, and det A > 0.
Transcribed Image Text:Problem 5: Let A € M2x2 (R). For u, v E R2, define
fA(u, v) = ut Av.
1
Here we are taking u and v to be 2 x 1 column vectors. Show that fA defines an inner
product on R2 if and only if A = A¹, a11 > 0, a22 > 0, and det A > 0.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
