Let A ∈ M2×2(R). For u, v ∈ R2, definefA(u, v) = utAv.(Here we are taking u and v to be 2 × 1 column vectors.) Show that fA defines aninner product on R2if and only if A = At, a11 > 0, a22 > 0, and det A > 0.
Let A ∈ M2×2(R). For u, v ∈ R2, definefA(u, v) = utAv.(Here we are taking u and v to be 2 × 1 column vectors.) Show that fA defines aninner product on R2if and only if A = At, a11 > 0, a22 > 0, and det A > 0.
Let A ∈ M2×2(R). For u, v ∈ R2, definefA(u, v) = utAv.(Here we are taking u and v to be 2 × 1 column vectors.) Show that fA defines aninner product on R2if and only if A = At, a11 > 0, a22 > 0, and det A > 0.
Let A ∈ M2×2(R). For u, v ∈ R 2 , define fA(u, v) = u tAv. (Here we are taking u and v to be 2 × 1 column vectors.) Show that fA defines an inner product on R 2 if and only if A = At , a11 > 0, a22 > 0, and det A > 0.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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