Question
If \(U\) and \(V\) are finite-dimensional vector space over \(F\). Then, \(U\) and \(V\) are isomorphic if and only if __________.
Answer
\(dim U = dim V\)
Question
\(T:U\rightarrow V\)is ________________ if and only if ker\(T = (0)\).
Answer
onto
Question
Suppose \(T:U\rightarrow V \)be the identity transformation, then \(R(I)\) is
Answer
V
Question
If \(T:U\rightarrow V\)be defined by \(T(u) = 0 \for all u \in U\), then \(T\) is a called ______transformation.
Answer
Zero
Question
The determinant of an \(m\times n\) matrix \(A\) is ____ to the rank of \(A\).
Answer
equal
Question
If \(A= diag (2,3,5)\) and \(B = diag(7,0,-4)\), then \(AB\) is ____________-
Answer
diag(14,0,-20)
Question
Suppose \(T\in A(V)\).Then is invertible if and only if the _______ term in the minimal polynomial of \(T\) is not zero.
Answer
constant
Question
If V be a vector space over Fof dimension n and \(T\in A(V)\) Then there exists a non-zero
polynomial Then there exists a non-zero polynomial Then there exists a non-zero polynomial \(g\) over F such that \(g(T) =\) ?
Answer
\(0\)
Question
The polynomial \(x^{2}+5x+6\) is of degree ________
Answer
2
Question
Given the vector space \(U\) over a field \(F\) , and the function \(T:U\rightarrow U \)defined by \(T(u) = u\), for all \(u\) belonging to \(U\).This transformation is called
Answer
identity
Question
The matrix \(0\) is called ____.
Answer
zero
Question
A and B are two matrices and \(AB=BA\), Then A and Bare said to be ___________.
Answer
Commutative
Question
A matrix obtained by subjecting Ito an elementary row or column operation is called a/an ___.
Answer
elementary
Question
If \(X\) and \(Y\) are matrices and \(XY =YX = I\). \(Y\) is said to be the ____ of \(A\).
Answer
inverse
Question
A square matrix \(A\) is said to be invertible if there exists a matrix B such that
Answer
\(AB=BA=I\)
Question
An ____________________ matrix is said to be of order \(n\).
Answer
\(mn\)
Question
A ________________________ is a matrix with the same number of rows and columns.
Answer
square
Question
A matrix with only one column is called a ___________ vector.
Answer
Column
Question
The pair of numbers and is called the ___________ of the matrix.
Answer
Size
Question
The number appearing in the various positions of a matrix are called the ________ of the matrix.
Answer
elements
Question
The __________ of \(T\) is defined to be the dimension of Kernel of \(T\).
Answer
Nullity
Question
The ______________ of a matrix is equal to the number of pivots in an echelon form of A
Answer
rank
Question
A matrix that does not have inverse is called a/an ________ matrix.
Answer
singular
Question
If X and Y are matrices and \(XY=YX=I\). Then, X is said to be
Answer
Non singular
Question
A matrix with only ___________ row is called a row matrix or row vector.
Answer
one
Question
Suppose \(A= m\) by nmatrix, the m-horizontal list of scalars are called _______________________
Answer
rows
Question
A rectangular array of scalars within square brackets is square is called
Answer
matrix
Question
Every vector space is isomorphic to its ______________ dual.
Answer
second
Question
If U be a vector space over F.Then, the space \(L(U,F)\) is called the __________ of \(U\).
Answer
dual space
Question
The dimension of \(R\) of \(T\) is called ______________ of \(T\).
Answer
rank
Question
If \(U\) and \(V\) are finite-dimensional vector space over \(F\). Then, \(U\) and \(V\) are isomorphic if and only if __________.
Answer
\(dim U = dim V\)
Question
\(T:U\rightarrow V\)is ________________ if and only if ker\(T = (0)\).
Answer
onto
Question
Suppose \(T:U\rightarrow V \)be the identity transformation, then \(R(I)\) is
Answer
V
Question
If \(T:U\rightarrow V\)be defined by \(T(u) = 0 \for all u \in U\), then \(T\) is a called ______transformation.
Answer
Zero
Question
The determinant of an \(m\times n\) matrix \(A\) is ____ to the rank of \(A\).
Answer
equal
Question
If \(A= diag (2,3,5)\) and \(B = diag(7,0,-4)\), then \(AB\) is ____________-
Answer
diag(14,0,-20)
Question
Suppose \(T\in A(V)\).Then is invertible if and only if the _______ term in the minimal polynomial of \(T\) is not zero.
Answer
constant
Question
If V be a vector space over Fof dimension n and \(T\in A(V)\) Then there exists a non-zero polynomial Then there exists a non-zero polynomial Then there exists a non-zero polynomial \(g\) over F such that \(g(T) =\) ?
Answer
\(0\)
Question
The polynomial \(x^{2}+5x+6\) is of degree ________
Answer
2
Question
Given the vector space \(U\) over a field \(F\) , and the function \(T:U\rightarrow U \)defined by \(T(u) = u\), for all \(u\) belonging to \(U\).This transformation is called
Answer
identity
Question
The matrix \(0\) is called ____.
Answer
zero
Question
A and B are two matrices and \(AB=BA\), Then A and Bare said to be ___________.
Answer
Commutative
Question
A matrix obtained by subjecting Ito an elementary row or column operation is called a/an ___.
Answer
elementary
Question
If \(X\) and \(Y\) are matrices and \(XY =YX = I\). \(Y\) is said to be the ____ of \(A\).
Answer
inverse
Question
A square matrix \(A\) is said to be invertible if there exists a matrix B such that
Answer
\(AB=BA=I\)
Question
An ____________________ matrix is said to be of order \(n\).
Answer
\(mn\)
Question
A ________________________ is a matrix with the same number of rows and columns.
Answer
square
Question
A matrix with only one column is called a ___________ vector.
Answer
Column
Question
The pair of numbers and is called the ___________ of the matrix.
Answer
Size
Question
The number appearing in the various positions of a matrix are called the ________ of the matrix.
Answer
elements
Question
The __________ of \(T\) is defined to be the dimension of Kernel of \(T\).
Answer
Nullity
Question
The ______________ of a matrix is equal to the number of pivots in an echelon form of A
Answer
rank
Question
A matrix that does not have inverse is called a/an ________ matrix.
Answer
singular
Question
If X and Y are matrices and \(XY=YX=I\). Then, X is said to be
Answer
Non singular
Question
A matrix with only ___________ row is called a row matrix or row vector.
Answer
one
Question
Suppose \(A= m\) by nmatrix, the m-horizontal list of scalars are called _______________________
Answer
rows
Question
A rectangular array of scalars within square brackets is square is called
Answer
matrix
Question
Every vector space is isomorphic to its ______________ dual.
Answer
second
Question
If U be a vector space over F.Then, the space \(L(U,F)\) is called the __________ of \(U\).
Answer
dual space
Question
The dimension of \(R\) of \(T\) is called ______________ of \(T\).
Answer
rank
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