Label the following statements as true or false.(a) Every quadratic form is a bilinear form.{b ) If two matrices are congruent, they have the same eigenvalues.(c) Symmetric bilinear forms have symmetric matrbc representations.(d) Any symmetric matrix is congruent to a diagonal matrix.(e) The sum of two symmetric bilinear forms is a symmetric bilinearform.(f) If two symmetric matrices over a field not of characteristic twohave the the same characteristic polynomial, then they are matrixrepresentations of the same bilinear form.(g) There exists a bilinear form H such that H (x, y) ≠ 0 for all x andy. (h ) If V is a vector space of dimension n, then dim(B(V)) = 2n.(i) Let H be a bilinear form on a finit~dimensional vector space Vwith dim(V) > 1. For any x ϵ V, t here exists y ϵ V such thaty ≠ 0, but H (x,y) = 0.(j) If H is any bilinear form on a finite-dimensional real inner productspace V, then there exists an ordered basis β for V such that vβ (H )is a diagonal matrix.
Label the following statements as true or false.
(a) Every quadratic form is a bilinear form.
{b ) If two matrices are congruent, they have the same eigenvalues.
(c) Symmetric bilinear forms have symmetric matrbc representations.
(d) Any symmetric matrix is congruent to a diagonal matrix.
(e) The sum of two symmetric bilinear forms is a symmetric bilinear
form.
(f) If two
have the the same characteristic polynomial, then they are matrix
representations of the same bilinear form.
(g) There exists a bilinear form H such that H (x, y) ≠ 0 for all x and
y.
(h ) If V is a
(i) Let H be a bilinear form on a finit~dimensional vector space V
with dim(V) > 1. For any x ϵ V, t here exists y ϵ V such that
y ≠ 0, but H (x,y) = 0.
(j) If H is any bilinear form on a finite-dimensional real inner product
space V, then there exists an ordered basis β for V such that vβ (H )
is a diagonal matrix.
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