List of top Mathematics Questions on Algebra

Let V be an inner product space and $S = \{\alpha_{1}, \alpha_{2}, \dots, \alpha_{m}\}$ be a finite subset of V. If S is an orthonormal set, then consider the following statements:
I: $||\alpha_{i}|| = 1$ for each $\alpha_{i} \in S$
II: $(\alpha_{i}, \alpha_{j}) = 0$ for $\alpha_{i}, \alpha_{j} \in S, i \neq j$.
Which of the following is correct?

If $\alpha, \beta$ are vectors in an inner product space V, then

For the vectors $u = (a,b), v = (c,d)$ in $C^{2}$ the inner product of $u$ and $v$ is defined by $\langle u,v \rangle = a\bar{c} + b\bar{d}$. If $u = (1+i, i), v = (i, 1-i)$ then $\langle u,v \rangle = $

Which of the following matrices is NOT diagonalizable?

The matrix $\left[\begin{matrix}1& 4\\ 3& 2\end{matrix}\right]$ is

If $\alpha = (-1,0,1), \beta = (2,0,-2) \in V_{3}(R)$ an inner product space, then $||\alpha + \beta|| = $

The necessary condition to diagonalize a matrix is that

For what values of a and b the following simultaneous equations have an infinite number of solutions? $x + y + z = 5, x + 3y + 3z = 9$, and $x + 2y + \alpha z = \beta$.

The characteristic equation associated with the matrix $\left[\begin{matrix}0& 0& 3\\ 1& 0& 2\\ 0& 1& 1\end{matrix}\right]$ is

The determinant of a skew-symmetric matrix of odd order is

The equation $\left|\begin{matrix}2& 1& 1\\ 1& 1&-1\\ y& x^{2}& x\end{matrix}\right|=0$ represents a parabola passing through the points}

If A and B are square matrices of size $n \times n$, then which of the following statement is not true.

Solution for the system of equations $4y + 3z = 8, 2x - z = 2$ and $3x + 2y = 5$ is

If $W=\{\left(\begin{matrix}x& y\\ z& 0\end{matrix}\right):x,y,z\in R\}$ is a subspace of the vector space $M_{2}$ of $2\times2$ matrices over the field of real numbers R, then $dimW=_$

Let $T:R^{3}\rightarrow R^{3}$ be a linear transformation defined by $T(a,b,c)=(a+b-c, a+b+c, b-c)$. Then the matrix of $T$ with respect to the ordered basis $B = \{(0,1,0), (0,0,1), (1,0,0)\}$ is}

If T is a linear transformation from $R^{2}\rightarrow R^{2}$ defined by $T(1,-1)=(2,-4)$ and $T(1,1)=(0,2)$. Then $T(a,b)=$

If $T:R^{3}\rightarrow R^{2}$ be a linear transformation defined by $T(a,b,c)=(a+b+c,a-b-c)$, then which of the following is an element in the null space of T.

The coordinates of (4,5,6) with respect to the basis set $e_{1}=(1,1,1), e_{2}=(-1,1,0), e_{3}=(1,0,-1)$ are}

Which of the following polynomial is irreducible over $Z_{3}$, a field modulo 3?

Let $A=\left[\begin{matrix}1& 1& 1\\ 2& 2& 3\\ x& y& 2\end{matrix}\right]$ and let $V=\{(x,y,z)\in R^{3}:\det(A)=0\}$. Then the dimension of V equals}

Let $f(x)=2+3x^{2}, g(x)=1+2x^{2}$ be polynomials in the ring $Z_{4}$ under mod 4. Then, $deg\{f(x)g(x)\}=$

Which of the following sets of vectors in $R^{2}$ are linearly independent over R.

The ring of integers (Z, +, $\cdot$) is

Let Z be the ring of integers and $f: Z \to 2Z$ defined by $f(n)=2n$, $\forall n \in Z$. Then f is

The ring of all $2 \times 2$ matrices over reals is