This is my matrix vector multiplication$-$

i want to see how the values get multiplied.. can you use pen and paper to show how the matrix and vector gets multiplied using the logic given below? Like how the row of one matrix and the column of the other matrix gets joined and multiplied? please work it out and show me $.$ Thank you so much

import org.apache.spark.sql$.$types.$\_$

import org.apache.spark.sql$.$functions.$\_$

$//$ Define matrix and vector data as sequences

val matrix$\_$xSeq $=$ Seq$($

$(0,0,1\mathrm{.}0),(0,1,2\mathrm{.}0),$

$(1,0,3\mathrm{.}0),(1,1,4\mathrm{.}0)$

$)$

val vector$\_$ySeq $=$ Seq$($

$(0,5\mathrm{.}0),$

$(1,6\mathrm{.}0),$

$)$

$//$ Convert matrix and vector data to DataFrames

val matrix$\_$xDF $=$ matrix$\_$xSeq.toDF$("$row$",$ "col", "val"$)$

val vector$\_$yDF $=$ vector$\_$ySeq.toDF$("$row$",$ "val"$)$

$//$ Show DataFrames

matrix$\_$xDF$.$show

vector$\_$yDF$.$show

matrix$\_$xDF$.$createOrReplaceTempView$("$X$")$

vector$\_$yDF$.$createOrReplaceTempView$("$Y$")$

val resultDF $=$ spark.sql$("""$

SELECT X$.$row, SUM$($X$.$val $*$ Y$.$val$)$ AS val

FROM X

JOIN Y ON X$.$col $=$ Y$.$row

GROUP BY X$.$row

$""")$

resultDF.show