WorksheetsLinear Algebra and Vector Spaces Worksheet
Total questions: 20
Worksheet time: 10mins
The rank nullity Theorem Applies to
Homogeneous differential equations
Linear transformation
Quadratic equation
skew symmetric matrices
The image of a linear transformation is
Null space
Range
Kernal
Domain
If nullity |T|=0 then
T is on to
T is one to one
T is not linear
T is zero transformation
If nullity = 0 then the transformation is
Not linear
injective
subjective
non - invertible
The rank of an identity matrix of order n is
0
n
n-1
1
A transformation is on to if
Rank = number of columns
Rank = number of rows
Nullity = number of columns
Nullity = number of rows
If v is an eigenvector of a matrix A and λ is the corresponding eigen value, which of the following is always true?
Av = v + λ
Av = λv
Av = v/λ
Av = 0
What is an eigenvalue of the matrix A = | 4 1 |
| 2 3 |
6
5
4
2
The matrix representation of a linear transformation depends on
Choice of bases
Choice of field only
Number of rows
Types of equations
If matrix A is 3 × 3 and nullity(A) = 1, then rank(A) =
0
1
2
3
The inner product of vectors u = [1,2], v = [3,4] in R^2 is
11
7
10
14
If u = (a, b), then ⟨u, u⟩ = 0 implies:
a = b
a = -b
a = 0, b = 0
a ≠ 0, b ≠ 0
If ||u|| = √⟨u, v⟩, Then ||u|| ≥ 0 is a property of:
Linear transformation
Inner product
Orthogonality
Basis
Which of the following is an orthonormal set in R^2?
[1,0], [0,1]
[1,1], [1,-1]
[2,0], [0,3]
[1,2], [2,1]
The dot product of two orthogonal vector is:
1
-1
0
Either 0 or 1
To make a vector v into a unit vector, divide it by:
Its magnitude
2
Its first coordinate
Its transpose
If u = (1/√2, 1/√2), then ||u|| =
1
0
√2
1/2
If →u = [3, 4], →v = [1, 0], then proj_→u_→v =
(3,0)
(0,4)
(1,1)
(2,2)
The projection vector lies:
Perpendicular to →v
Along →v
Along →u
In a random direction
In Gram-Schmidt process the projection is used for
Normalize vectors
Make vector orthogonal
From a determinant
None
