Let aaa be a real parameter and take these four vectors in R4\mathbb{R}^4R4:
v1=(1021),v2=(2153),v3=(0111),v4=(113a)v_1=\begin{pmatrix}1\\0\\2\\1\end{pmatrix},\qquad v_2=\begin{pmatrix}2\\1\\5\\3\end{pmatrix},\qquad v_3=\begin{pmatrix}0\\1\\1\\1\end{pmatrix},\qquad v_4=\begin{pmatrix}1\\1\\3\\a\end{pmatrix}v1=1021,v2=2153,v3=0111,v4=113a
Let M=( v1 v2 v3 v4 )M = \big(\,v_1\;\;v_2\;\;v_3\;\;v_4\,\big)M=(v1v2v3v4) be the 4×44\times44×4 matrix whose columns are these vectors.
Give rank(M)\operatorname{rank}(M)rank(M) and the nullity dimker(M)\dim\ker(M)dimker(M) for a=2a = 2a=2 and for a=5a = 5a=5.
Select all that apply.
a=2a=2a=2: rank 333, nullity 111. a=5a=5a=5: rank 444, nullity 000.
a=2a=2a=2: rank 333, nullity 111. a=5a=5a=5: rank 333, nullity 111.
a=2a=2a=2: rank 222, nullity 222. a=5a=5a=5: rank 333, nullity 111.
a=2a=2a=2: rank 222, nullity 222. a=5a=5a=5: rank 444, nullity 000.