QUESTIONS - LINEAR ALGEBRA - VECTORS
QUESTIONS / ANSWERS .ACF.LINEAR ALGEBRA
On 2/9/07, acf
> acf has left a new comment on your post "QUESTIONS - LINEAR ALGEBRA &
> VECTORS":
>
> Let B and C be non-parallel vectors in R superscript 3 and let A be an
> element in R superscript 3 orthgonal to both B and C. Use liner algebra to
> prove A is parallel to B cross C.
>
> Hint: Find a basis. Please do not use vector compounents or simple vector
> algebra. Must use linear algebra techniques.
>
>
>
> Posted by acf to Maths,management,counselling at 11:50 AM
R3 DIMENSION.
B AND C ARE NON PARALLEL.
HENCE THEY ARE INDEPENDENT VECTORS
A IS ORTHOGONAL TO BOTH B AND C.
HENCE A IS NEITHER DEPENDENT ON B NOR ON C.
HENCE A,B,C ARE 3 INDEPENDENT VECTORS IN R3
HENCE A,B,C FORM A BASIS IN R3
HENCE
B X C = xA+yB+zC
BUT
B X C IS PERPENDICULAR TO BOTH B AND C.
HENCE IT CAN NOT HAVE A COMPONENT ALONG B OR C...THAT IS ITS COMPONENTS ALONG B AND C ARE ZEROS
THAT IS
y=z=0
HENCE
B X C = xA
HENCE
B X C AND A ARE DEPENDENT ....COLLINEAR OR PARALLEL VECTORS.
posted by mathsmanagementcommonsense at 8:35 AM
5 Comments:
question:Determine whether the following transformation is a linear transformation:
T:R^2>>R^3 where
T([x,y]^t)=[ln(x),x+y]^t
For all [x,y]^t such that ln(x) is defined.
Why or why not? Be very clear in terms of the important properties of linear transformations!!
11:47 PM
Question: Find a matrix
A which reflects every vector[x,y]^t symmetrically about the origin in R^2. Show this by making up some vector in R^2 and applying this transformation. Finally, graph both vectors in R^2.
11:49 PM
Question:3: Define a transformation T:R^2>>>R^2 by the following rule:T(X) is the result offirst "squashing"[1 0] and [0 4] then rotating X counterclockwise by pi/3radians.
a)Describe clearly in words the image of the unit circle x^2+ y^2=1 under this transformation.
b)If this new image were to be graphed in the w_1w_2 plane, what would be its algebraic equation?
11:55 PM
(solve the vibrating membrane problem:
d^2u/dt^2 = c^2(d^2u/dr^2+1/r*du/dr), where u = u(r,t), 1>r>0, c=1, and t>0
boundary conditions:
u(a,t) = 0, for all t>=0
initial conditions:
u(r,0) = f(r), du/dt(r,0) = g(r), 1>r>0
where, a = 1, g(r) = 1-r^2, f(r) = J_0(alpha_3*r)
where J_0 is the bessel function of order zero and of the first kind; alpha_3 represents the third root of the bessel function J_0(x))
6:43 PM
can someone help me>>?
Let S, T, U, and V be sets, and let F : S→ T, G1 : T → U, G2 : T →U, and
H : U→ V be functions.
a) Suppose that H ο G1 = H ο G2 . Show that if H is one-to-one, then G1 = G2. (Hint: To show that two functions with the same domain and co-domain are equal, it suffices to show that they do the same thing to each element of their domain. That is, to show that G1 = G2, show that G1(t) = G2(t) for all tεT.)
b)Suppose that H ο G1 = H ο G2. Show by counterexample that if h is not one-to-one, then it is not necessarily true that G1 = G2.
c) Suppose that G1 ο F = G2 ο F. Show that if F is onto, then G1= G2
d). Suppose that G1 ο F = G2 ο F. Show by counterexample that if F is not onto, then it is not necessarily true that G1= G2
10:15 AM
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