Linear algebra key points
Linear Algebra for Communication (HNECT409)
Model Question Paper – Answer Key
PART A
1. A 3×5 matrix with rank 3 has full row rank. Therefore Ax=b is consistent for every b. Since there
are 5 unknowns and rank 3, there are 2 free variables, so infinitely many solutions exist.
2. A subspace is a subset closed under vector addition and scalar multiplication and containing the
zero vector.
3. Algebraic multiplicity: multiplicity as a root of characteristic polynomial. Geometric multiplicity:
dimension of eigenspace.
4. For a triangular matrix, det(A−λI) equals the product of diagonal terms, so eigenvalues are the
diagonal entries.
5. A vector orthogonal to [1,2,1] and [2,5,4] is (3,−2,1).
6. If u⊥v, then ||u+v||² = ||u||² + ||v||².
7. A quadratic form x■Ax is positive definite if x■Ax > 0 for all non-zero x.
8. SVD: A = UΣV■ where U and V are orthogonal matrices and Σ contains singular values.
PART B
9(a). Determinant of [[1,4,2],[2,5,1],[3,6,0]] = 9 ≠ 0. Hence the vectors are linearly independent and
form a basis.
9(b). Null(C) = { t[1,−1]■ : t ∈ R }.
10(a). The first quadrant is not a subspace because it is not closed under multiplication by negative
scalars.
10(b). 3t²+5t−5 = (52/7)p■ − (27/7)p■ + (23/7)p■.
11(a). Reflection about x-axis: A=[[1,0],[0,−1]]. Eigenvalues: 1 and −1. Also A²=I.
11(b). A matrix is diagonalizable iff it possesses n linearly independent eigenvectors.
12(a). Eigenvalues: 1±i. Corresponding eigenvectors: [1,i]■ and [1,−i]■.
12(b). If Ax=λx and A is real, taking conjugates gives Ax■ = λ■x■.
13(a). Projection matrix onto W with orthogonal basis {u■}: P = Σ(u■u■■)/(u■■u■).
13(b). Orthogonal basis obtained by Gram–Schmidt: {(3,6,0),(0,0,2)}.
14(a). For orthogonal Q, Q■Q=I, hence norms and angles are preserved.
14(b). Least-squares solutions satisfy the normal equations A■Ax=A■b.
15(a). Spectral decomposition of symmetric A: A=PDP■ = Σ λ■u■u■■.
15(b). Properties: real eigenvalues, orthogonal eigenvectors, orthogonal diagonalization, A=A■,
complete orthonormal eigenbasis.
16(a). Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal.
16(b). x■Ax is maximized at the eigenvector corresponding to the largest eigenvalue.
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