MATH225 EXAM 2026 LINEAR ALGEBRA
MATRICES SYSTEMS STUDY GUIDE
GRADED A+
◉ Two vectors are orthogonal if.
Answer: u(dot)v = 0
◉ v•v=||v||^2.
Answer: The given statement is true. By the definition of the length
of a vector v, ||v||=sqrt(v•v)
◉ For any scalar c, u•(cv)=c(u•v)..
Answer: The given statement is true because this is a valid property
of the inner product.
◉ If the distance from u to v equals the distance from u to −v, then u
and v are orthogonal..
Answer: By the definition of orthogonal, u and v are orthogonal if
and only if u•v=0. This happens if and only if 2u*v = -2u*v, which
happens if and only if the squared distance from u to v equals the
squared distance from u to −v. Requiring the squared distances to be
equal is the same as requiring the distances to be equal, so the given
statement is true.
, ◉ For a square matrix A, vectors in Col A are orthogonal to vectors
in Nul A..
Answer: The given statement is false. By the theorem of orthogonal
complements, it is known that vectors in
Col A are orthogonal to vectors in Nul AT. Using the definition of
orthogonal complements, vectors in Col A
are orthogonal to vectors in Nul A if and only if the rows and
columns of A are the same, which is not necessarily true.
◉ If vectors v1,...,vp span a subspace W and if x is orthogonal to each
vj for j=1,...,p, then x is in W⊥.
Answer: The given statement is true. If x is orthogonal to each vj,
then x is also orthogonal to any linear combination of those vj. Since
any vector in W can be described as a linear combination of vj, x is
orthogonal to all vectors in W.
◉ u•v−v•u=0.
Answer: The given statement is true. Since the inner product is
commutative, u•v=v•u. Subtracting v•u from each side of this
equation gives u•v−v•u=0.
◉ For any scalar c, ||cv||=c||v||..
MATRICES SYSTEMS STUDY GUIDE
GRADED A+
◉ Two vectors are orthogonal if.
Answer: u(dot)v = 0
◉ v•v=||v||^2.
Answer: The given statement is true. By the definition of the length
of a vector v, ||v||=sqrt(v•v)
◉ For any scalar c, u•(cv)=c(u•v)..
Answer: The given statement is true because this is a valid property
of the inner product.
◉ If the distance from u to v equals the distance from u to −v, then u
and v are orthogonal..
Answer: By the definition of orthogonal, u and v are orthogonal if
and only if u•v=0. This happens if and only if 2u*v = -2u*v, which
happens if and only if the squared distance from u to v equals the
squared distance from u to −v. Requiring the squared distances to be
equal is the same as requiring the distances to be equal, so the given
statement is true.
, ◉ For a square matrix A, vectors in Col A are orthogonal to vectors
in Nul A..
Answer: The given statement is false. By the theorem of orthogonal
complements, it is known that vectors in
Col A are orthogonal to vectors in Nul AT. Using the definition of
orthogonal complements, vectors in Col A
are orthogonal to vectors in Nul A if and only if the rows and
columns of A are the same, which is not necessarily true.
◉ If vectors v1,...,vp span a subspace W and if x is orthogonal to each
vj for j=1,...,p, then x is in W⊥.
Answer: The given statement is true. If x is orthogonal to each vj,
then x is also orthogonal to any linear combination of those vj. Since
any vector in W can be described as a linear combination of vj, x is
orthogonal to all vectors in W.
◉ u•v−v•u=0.
Answer: The given statement is true. Since the inner product is
commutative, u•v=v•u. Subtracting v•u from each side of this
equation gives u•v−v•u=0.
◉ For any scalar c, ||cv||=c||v||..
